import { type PracticeQuestion } from "../../types/lesson";

export const LINEAR_EQ_TWO_VAR_EASY: PracticeQuestion[] = [
  {
    id: "029c2dc2",
    type: "mcq",
    questionHtml:
      "A teacher is creating an assignment worth <strong>70</strong> points. The assignment will consist of questions worth <strong>1</strong> point and questions worth <strong>3</strong> points. Which equation represents this situation, where <strong>x</strong> represents the number of <strong>1</strong>-point questions and <strong>y</strong> represents the number of <strong>3</strong>-point questions?",
    choices: [
      { label: "A", text: "<strong>4 xy = 70</strong>" },
      { label: "B", text: "<strong>4 (x + y) = 70</strong>" },
      { label: "C", text: "<strong>3 x + y = 70</strong>" },
      { label: "D", text: "<strong>x + 3 y = 70</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Since <strong>x</strong> represents the number of <strong>1</strong>-point questions and <strong>y</strong> represents the number of <strong>3</strong>-point questions, the assignment is worth a total of <strong>1 dot x + 3 dot y</strong>, or <strong>x + 3 y</strong>, points. Since the assignment is worth <strong>70</strong> points, the equation <strong>x + 3 y = 70</strong> represents this situation.<br>Choice A is incorrect and may result from conceptual errors.<br>Choice B is incorrect and may result from conceptual errors.<br>Choice C is incorrect and may result from conceptual errors.",
    hasFigure: false,
  },
  {
    id: "10c448d6",
    type: "mcq",
    questionHtml:
      "A line in the xy-plane has a slope of <strong>one ninth</strong> and passes through the point <strong>(0, 14)</strong>. Which equation represents this line?",
    choices: [
      { label: "A", text: "<strong>y = −one ninth x − 14</strong>" },
      { label: "B", text: "<strong>y = −one ninth x + 14</strong>" },
      { label: "C", text: "<strong>y = one ninth x − 14</strong>" },
      { label: "D", text: "<strong>y = one ninth x + 14</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The equation of a line in the xy-plane can be written as <strong>y = m x + b</strong>, where <strong>m</strong> represents the slope of the line and <strong>(0, b)</strong> represents the y-intercept of the line. It's given that the slope of the line is <strong>one ninth</strong>. It follows that <strong>m = one ninth</strong>. It's also given that the line passes through the point <strong>(0, 14)</strong>. It follows that <strong>b = 14</strong>. Substituting <strong>one ninth</strong> for <strong>m</strong> and <strong>14</strong> for <strong>b</strong> in <strong>y = m x + b</strong> yields <strong>y = one ninth x + 14</strong>. Thus, the equation <strong>y = one ninth x + 14</strong> represents this line.<br>Choice A is incorrect. This equation represents a line with a slope of <strong>−one ninth</strong> and a y-intercept of <strong>(0 −14)</strong>.<br>Choice B is incorrect. This equation represents a line with a slope of <strong>−one ninth</strong> and a y-intercept of <strong>(0, 14)</strong>.<br>Choice C is incorrect. This equation represents a line with a slope of <strong>one ninth</strong> and a y-intercept of <strong>(0 −14)</strong>.",
    hasFigure: false,
  },
  {
    id: "12ae3452",
    type: "spr",
    questionHtml:
      "The equation <strong>46 = 2 a + 2 b</strong> gives the relationship between the side lengths <strong>a</strong> and <strong>b</strong> of a certain parallelogram. If <strong>a = 9</strong>, what is the value of <strong>b</strong>?",
    choices: [],
    correctAnswer: "14",
    explanation:
      "The correct answer is <strong>14</strong>. It's given that the equation <strong>46 = 2 a + 2 b</strong> gives the relationship between the side lengths <strong>a</strong> and <strong>b</strong> of a certain parallelogram. Substituting <strong>9</strong> for <strong>a</strong> in the given equation yields <strong>46 = 2 (9) + 2 b</strong>, or <strong>46 = 18 + 2 b</strong>. Subtracting <strong>18</strong> from both sides of this equation yields <strong>28 = 2 b</strong>. Dividing both sides of this equation by <strong>2</strong> yields <strong>14 = b</strong>. Therefore, if <strong>a = 9</strong>, the value of <strong>b</strong> is <strong>14</strong>.",
    hasFigure: false,
  },
  {
    id: "174885f8",
    type: "mcq",
    questionHtml:
      "Jay walks at a speed of <strong>3</strong> miles per hour and runs at a speed of <strong>5</strong> miles per hour. He walks for <strong>w</strong> hours and runs for <strong>r</strong> hours for a combined total of <strong>14</strong> miles. Which equation represents this situation?",
    choices: [
      { label: "A", text: "<strong>3 w + 5 r = 14</strong>" },
      { label: "B", text: "<strong>one third w + one fifth r = 14</strong>" },
      { label: "C", text: "<strong>one third w + one fifth r = 112</strong>" },
      { label: "D", text: "<strong>3 w + 5 r = 112</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Since Jay walks at a speed of <strong>3</strong> miles per hour for <strong>w</strong> hours, Jay walks a total of <strong>3 w</strong> miles. Since Jay runs at a speed of <strong>5</strong> miles per hour for <strong>r</strong> hours, Jay runs a total of <strong>5 r</strong> miles. Therefore, the total number of miles Jay travels can be represented by <strong>3 w + 5 r</strong>. Since the combined total number of miles is <strong>14</strong>, the equation <strong>3 w + 5 r = 14</strong> represents this situation.<br>Choice B is incorrect and may result from conceptual errors.<br>Choice C is incorrect and may result from conceptual errors.<br>Choice D is incorrect and may result from conceptual errors.",
    hasFigure: false,
  },
  {
    id: "1efd8202",
    type: "mcq",
    questionHtml:
      "<strong>y = 70 x + 8</strong><br>Which table gives three values of <strong>x</strong> and their corresponding values of <strong>y</strong> for the given equation?",
    choices: [
      {
        label: "A",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>0</strong><br><strong>8</strong><br><br><strong>2</strong><br><strong>148</strong><br><br><strong>4</strong><br><strong>288</strong>",
      },
      {
        label: "B",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>0</strong><br><strong>70</strong><br><br><strong>2</strong><br><strong>78</strong><br><br><strong>4</strong><br><strong>86</strong>",
      },
      {
        label: "C",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>0</strong><br><strong>70</strong><br><br><strong>2</strong><br><strong>140</strong><br><br><strong>4</strong><br><strong>280</strong>",
      },
      {
        label: "D",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>0</strong><br><strong>8</strong><br><br><strong>2</strong><br><strong>132</strong><br><br><strong>4</strong><br><strong>272</strong>",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Each of the given choices gives three values of <strong>x</strong>: <strong>0</strong>, <strong>2</strong>, and <strong>4</strong>. Substituting <strong>0</strong> for <strong>x</strong> in the given equation yields <strong>y = 70 (0) + 8</strong>, or <strong>y = 8</strong>. Therefore, when <strong>x = 0</strong>, the corresponding value of <strong>y</strong> for the given equation is <strong>8</strong>. Substituting <strong>2</strong> for <strong>x</strong> in the given equation yields <strong>y = 70 (2) + 8</strong>, or <strong>y = 148</strong>. Therefore, when <strong>x = 2</strong>, the corresponding value of <strong>y</strong> for the given equation is <strong>148</strong>. Substituting <strong>4</strong> for <strong>x</strong> in the given equation yields <strong>y = 70 (4) + 8</strong>, or <strong>y = 288</strong>. Therefore, when <strong>x = 4</strong>, the corresponding value of <strong>y</strong> for the given equation is <strong>288</strong>. Thus, if the three values of <strong>x</strong> are <strong>0</strong>, <strong>2</strong>, and <strong>4</strong>, then their corresponding values of <strong>y</strong> are <strong>8</strong>, <strong>148</strong>, and <strong>288</strong>, respectively, for the given equation.<br>Choice B is incorrect. This table gives three values of <strong>x</strong> and their corresponding values of <strong>y</strong> for the equation <strong>y = 4 x + 70</strong>.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "24854644",
    type: "mcq",
    questionHtml:
      "What is the equation of the line that passes through the point <strong>(0, 5)</strong> and is parallel to the graph of <strong>y = 7 x + 4</strong> in the xy-plane?",
    choices: [
      { label: "A", text: "<strong>y = 5 x</strong>" },
      { label: "B", text: "<strong>y = 7 x + 5</strong>" },
      { label: "C", text: "<strong>y = 7 x</strong>" },
      { label: "D", text: "<strong>y = 5 x + 7</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The equation of a line in the xy-plane can be written in slope-intercept form <strong>y = m x + b</strong>, where <strong>m</strong> is the slope of the line and <strong>(0, b)</strong> is its y-intercept. It’s given that the line passes through the point <strong>(0, 5)</strong>. Therefore, <strong>b = 5</strong>. It’s also given that the line is parallel to the graph of <strong>y = 7 x + 4</strong>, which means the line has the same slope as the graph of <strong>y = 7 x + 4</strong>. The slope of the graph of <strong>y = 7 x + 4</strong> is <strong>7</strong>. Therefore, <strong>m = 7</strong>. Substituting <strong>7</strong> for <strong>m</strong> and <strong>5</strong> for <strong>b</strong> in the equation <strong>y = m x + b</strong> yields <strong>y = 7 x + 5</strong>.<br>Choice A is incorrect. The graph of this equation passes through the point <strong>(0, 0)</strong>, not <strong>(0, 5)</strong>, and has a slope of <strong>5</strong>, not <strong>7</strong>.<br>Choice C is incorrect. The graph of this equation passes through the point <strong>(0, 0)</strong>, not <strong>(0, 5)</strong>.<br>Choice D is incorrect. The graph of this equation passes through the point <strong>(0, 7)</strong>, not <strong>(0, 5)</strong>, and has a slope of <strong>5</strong>, not <strong>7</strong>.",
    hasFigure: false,
  },
  {
    id: "2554b413",
    type: "mcq",
    questionHtml:
      "In the xy-plane, a line has a slope of 6 and passes through the point <strong>with coordinates zero, 8</strong>. Which of the following is an equation of this line?",
    choices: [
      { label: "A", text: "<strong>y = 6 x + 8</strong>" },
      { label: "B", text: "<strong>y = 6 x + 48</strong>" },
      { label: "C", text: "<strong>y = 8 x + 6</strong>" },
      { label: "D", text: "<strong>y = 8 x + 48</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The slope-intercept form of an equation for a line is <strong>y = m x + b</strong>, where m is the slope of the line and b is the y-coordinate of the y-intercept of the line. It’s given that the slope is 6, so <strong>m = 6</strong>. It’s also given that the line passes through the point <strong>with coordinates 0, 8</strong> on the y-axis, so <strong>b = 8</strong>. Substituting <strong>m = 6</strong> and <strong>b = 8</strong> into the equation <strong>y = m x + b</strong> gives <strong>y = 6 x + 8</strong>.Choices B, C, and D are incorrect and may result from misinterpreting the slope-intercept form of an equation of a line.",
    hasFigure: false,
  },
  {
    id: "2d0e13a6",
    type: "spr",
    questionHtml:
      "Line <strong>k</strong> is defined by <strong>y = one fourth x + 1</strong>. Line <strong>j</strong> is parallel to line <strong>k</strong> in the xy-plane. What is the slope of <strong>j</strong>?",
    choices: [],
    correctAnswer: ".25, 1/4",
    explanation:
      "The correct answer is <strong>one fourth</strong>. It's given that line <strong>k</strong> is defined by <strong>y = one fourth x + 1</strong>. It's also given that line <strong>j</strong> is parallel to line <strong>k</strong> in the xy-plane. A line in the xy-plane represented by an equation in slope-intercept form <strong>y = m x + b</strong> has a slope of <strong>m</strong> and a y-intercept of <strong>(0, b)</strong>. Therefore, the slope of line <strong>k</strong> is <strong>one fourth</strong>. Since parallel lines have equal slopes, the slope of line <strong>j</strong> is <strong>one fourth</strong>. Note that 1/4 and .25 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "4acd05cd",
    type: "mcq",
    questionHtml:
      "The line slants sharply up from left to right.<br>The line passes through the following points:<br><br>(0 comma negative 5)<br>(1 comma negative 3)<br>(2 comma negative 1)<br><br>The graph shows the linear relationship between <strong>x</strong> and <strong>y</strong>. Which table gives three values of <strong>x</strong> and their corresponding values of <strong>y</strong> for this relationship?",
    choices: [
      {
        label: "A",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>0</strong><br><strong>0</strong><br><br><strong>1</strong><br><strong>−7</strong><br><br><strong>2</strong><br><strong>−9</strong>",
      },
      {
        label: "B",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>0</strong><br><strong>0</strong><br><br><strong>1</strong><br><strong>−3</strong><br><br><strong>2</strong><br><strong>−1</strong>",
      },
      {
        label: "C",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>0</strong><br><strong>−5</strong><br><br><strong>1</strong><br><strong>−7</strong><br><br><strong>2</strong><br><strong>−9</strong>",
      },
      {
        label: "D",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>0</strong><br><strong>−5</strong><br><br><strong>1</strong><br><strong>−3</strong><br><br><strong>2</strong><br><strong>−1</strong>",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It’s given that the graph shows the linear relationship between <strong>x</strong> and <strong>y</strong>. The given graph passes through the points <strong>(0 −5)</strong>, <strong>(1 −3)</strong>, and <strong>(2 −1)</strong>. It follows that when <strong>x = 0</strong>, the corresponding value of <strong>y</strong> is <strong>−5</strong>, when <strong>x = 1</strong>, the corresponding value of <strong>y</strong> is <strong>−3</strong>, and when <strong>x = 2</strong>, the corresponding value of <strong>y</strong> is <strong>−1</strong>. Of the given choices, only the table in choice D gives these three values of <strong>x</strong> and their corresponding values of <strong>y</strong> for the relationship shown in the graph.<br>Choice A is incorrect. This table represents a relationship between <strong>x</strong> and <strong>y</strong> such that the graph passes through the points <strong>(0, 0)</strong>, <strong>(1 −7)</strong>, and <strong>(2 −9)</strong>.<br>Choice B is incorrect. This table represents a relationship between <strong>x</strong> and <strong>y</strong> such that the graph passes through the points <strong>(0, 0)</strong>, <strong>(1 −3)</strong>, and <strong>(2 −1)</strong>.<br>Choice C is incorrect. This table represents a linear relationship between <strong>x</strong> and <strong>y</strong> such that the graph passes through the points <strong>(0 −5)</strong>, <strong>(1 −7)</strong>, and <strong>(2 −9)</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/4acd05cd_svg1.svg",
  },
  {
    id: "520e6f5b",
    type: "mcq",
    questionHtml:
      "A city is planning to build a rock retaining wall, a monument, and a garden in a park. The table above shows four rock types that will be considered for use in the project. Also shown for each rock type is its weight per volume, in pounds per cubic foot (lb/ft3), and the cost per pound, in dollars. The equation <strong>0 . 0 3 · (120 w, ) + 0 . 1 8 · (180 z, ) + 3, 385 . 8 0 = 7, 576 . 2 0.</strong> gives the total cost, in dollars, of the rocks used in the project in terms of the number of ft3 of limestone, w, and the number of ft3 of basalt, z. All four rock types are used in the project. Which of the following is the best interpretation of 3,385.80 in this context?",
    choices: [
      {
        label: "A",
        text: "The cost of the granite and sandstone needed for the project",
      },
      {
        label: "B",
        text: "The cost of the basalt and limestone needed for the project",
      },
      { label: "C", text: "The cost of the basalt needed for the project" },
      { label: "D", text: "The cost of the sandstone needed for the project" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The table shows the cost of limestone is $0.03/lb, and the weight per volume for limestone is 120 lb/ft3. Therefore, the term <strong>0 . 0 3 · 120 w</strong> represents the cost, in dollars, of w ft3 of limestone. Similarly, the term <strong>0 . 1 8 · 180 z</strong> represents the cost, in dollars, of z ft3 of basalt. The given equation shows that the total cost of all the rocks used in the project is $7,576.20. Since it’s given that all four rock types are used in the project, the remaining term, 3,385.80, represents the cost, in dollars, of the granite and sandstone needed for the project.Choice B is incorrect. The cost of basalt and limestone needed for the project can be represented by <strong>0 . 1 8 · 180 z + 0 . 0 3 · 120 w</strong>. Choice C is incorrect. The cost of the basalt needed for the project can be represented by the expression <strong>0 . 1 8 · 180 z</strong>. Choice D is incorrect and may result from neglecting to include granite in the rock types used for the project.",
    hasFigure: false,
  },
  {
    id: "52a8ef85",
    type: "mcq",
    questionHtml:
      "The equation <strong>40 x + 20 y = 160</strong> represents the number of sweaters, <strong>x</strong>, and number of shirts, <strong>y</strong>, that Yesenia purchased for <strong>dollar sign 160</strong>. If Yesenia purchased <strong>2</strong> sweaters, how many shirts did she purchase?",
    choices: [
      { label: "A", text: "<strong>3</strong>" },
      { label: "B", text: "<strong>4</strong>" },
      { label: "C", text: "<strong>8</strong>" },
      { label: "D", text: "<strong>40</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It's given that the equation <strong>40 x + 20 y = 160</strong> represents the number of sweaters, <strong>x</strong>, and the number of shirts, <strong>y</strong>, that Yesenia purchased for <strong>dollar sign 160</strong>. If Yesenia purchased <strong>2</strong> sweaters, the number of shirts she purchased can be calculated by substituting <strong>2</strong> for <strong>x</strong> in the given equation, which yields <strong>40 (2) + 20 y = 160</strong>, or <strong>80 + 20 y = 160</strong>. Subtracting <strong>80</strong> from both sides of this equation yields <strong>20 y = 80</strong>. Dividing both sides of this equation by <strong>20</strong> yields <strong>y = 4</strong>. Therefore, if Yesenia purchased <strong>2</strong> sweaters, she purchased <strong>4</strong> shirts.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect. This is the number of shirts Yesenia purchased if she purchased <strong>0</strong> sweaters.<br>Choice D is incorrect. This is the price, in dollars, for each sweater, not the number of shirts Yesenia purchased.",
    hasFigure: false,
  },
  {
    id: "535fa6e6",
    type: "mcq",
    questionHtml:
      "Davio bought some potatoes and celery. The potatoes cost <strong>dollar sign 0.69</strong> per pound, and the celery cost <strong>dollar sign 0.99</strong> per pound. If Davio spent <strong>dollar sign 5.34</strong> in total and bought twice as many pounds of celery as pounds of potatoes, how many pounds of celery did Davio buy?",
    choices: [
      { label: "A", text: "<strong>2</strong>" },
      { label: "B", text: "<strong>2.5</strong>" },
      { label: "C", text: "<strong>2.67</strong>" },
      { label: "D", text: "<strong>4</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Let <strong>p</strong> represent the number of pounds of potatoes and let <strong>c</strong> represent the number of pounds of celery that Davio bought. It’s given that potatoes cost <strong>dollar sign 0.69</strong> per pound and celery costs <strong>dollar sign 0.99</strong> per pound. If Davio spent <strong>dollar sign 5.34</strong> in total, then the equation <strong>0.69 p + 0.99 c = 5.34</strong> represents this situation. It’s also given that Davio bought twice as many pounds of celery as pounds of potatoes; therefore, <strong>c = 2 p</strong>. Substituting <strong>2 p</strong> for <strong>c</strong> in the equation <strong>0.69 p + 0.99 c = 5.34</strong> yields <strong>0.69 p + 0.99 (2 p) = 5.34</strong>, which is equivalent to <strong>0.69 p + 1.98 p = 5.34</strong>, or <strong>2.67 p = 5.34</strong>. Dividing both sides of this equation by <strong>2.67</strong> yields <strong>p = 2</strong>. Substituting <strong>2</strong> for <strong>p</strong> in the equation <strong>c = 2 p</strong> yields <strong>c = 2 (2)</strong>, or <strong>c = 4</strong>. Therefore, Davio bought <strong>4</strong> pounds of celery.<br>Choice A is incorrect. This is the number of pounds of potatoes, not the number of pounds of celery, Davio bought.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "5b8a8475",
    type: "mcq",
    questionHtml:
      "Line <strong>k</strong> is defined by <strong>y = 3 x + 15</strong>. Line <strong>j</strong> is perpendicular to line <strong>k</strong> in the xy-plane. What is the slope of line <strong>j</strong>?",
    choices: [
      { label: "A", text: "<strong>−one third</strong>" },
      { label: "B", text: "<strong>−one twelfth</strong>" },
      { label: "C", text: "<strong>−one eighteenth</strong>" },
      { label: "D", text: "<strong>−one forty fifth</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It's given that line <strong>j</strong> is perpendicular to line <strong>k</strong> in the xy-plane. It follows that the slope of line <strong>j</strong> is the opposite reciprocal of the slope of line <strong>k</strong>. The equation for line <strong>k</strong> is written in slope-intercept form <strong>y = m x + b</strong>, where <strong>m</strong> is the slope of the line and <strong>b</strong> is the y-coordinate of the y-intercept of the line. It follows that the slope of line <strong>k</strong> is <strong>3</strong>. The opposite reciprocal of a number is <strong>−1</strong> divided by the number. Thus, the opposite reciprocal of <strong>3</strong> is <strong>−one third</strong>. Therefore, the slope of line <strong>j</strong> is <strong>−one third</strong>.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "7038b587",
    type: "spr",
    questionHtml:
      "Vivian bought party hats and cupcakes for <strong>dollar sign 71</strong>. Each package of party hats cost <strong>dollar sign 3</strong>, and each cupcake cost <strong>dollar sign 1</strong>. If Vivian bought <strong>10</strong> packages of party hats, how many cupcakes did she buy?",
    choices: [],
    correctAnswer: "41",
    explanation:
      "The correct answer is <strong>41</strong>. The number of cupcakes Vivian bought can be found by first finding the amount Vivian spent on cupcakes. The amount Vivian spent on cupcakes can be found by subtracting the amount Vivian spent on party hats from the total amount Vivian spent. The amount Vivian spent on party hats can be found by multiplying the cost per package of party hats by the number of packages of party hats, which yields <strong>dollar sign 3 dot 10</strong>, or <strong>dollar sign 30</strong>. Subtracting the amount Vivian spent on party hats, <strong>dollar sign 30</strong>, from the total amount Vivian spent, <strong>dollar sign 71</strong>, yields <strong>dollar sign 71 − dollar sign 30</strong>, or <strong>dollar sign 41</strong>. Since the amount Vivian spent on cupcakes was <strong>dollar sign 41</strong> and each cupcake cost <strong>dollar sign 1</strong>, it follows that Vivian bought <strong>41</strong> cupcakes.",
    hasFigure: false,
  },
  {
    id: "768b2425",
    type: "mcq",
    questionHtml:
      "Last week, an interior designer earned a total of <strong>dollar sign 1, 258</strong> from consulting for <strong>x</strong> hours and drawing up plans for <strong>y</strong> hours. The equation <strong>68 x + 85 y = 1, 258</strong> represents this situation. Which of the following is the best interpretation of <strong>68</strong> in this context?",
    choices: [
      {
        label: "A",
        text: "The interior designer earned <strong>dollar sign 68</strong> per hour consulting last week.",
      },
      {
        label: "B",
        text: "The interior designer worked <strong>68</strong> hours drawing up plans last week.",
      },
      {
        label: "C",
        text: "The interior designer earned <strong>dollar sign 68</strong> per hour drawing up plans last week.",
      },
      {
        label: "D",
        text: "The interior designer worked <strong>68</strong> hours consulting last week.",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It's given that <strong>68 x + 85 y = 1, 258</strong> represents the situation where an interior designer earned a total of <strong>dollar sign 1, 258</strong> last week from consulting for <strong>x</strong> hours and drawing up plans for <strong>y</strong> hours. Thus, <strong>68 x</strong> represents the amount earned, in dollars, from consulting for <strong>x</strong> hours, and <strong>85 y</strong> represents the amount earned, in dollars, from drawing up plans for <strong>y</strong> hours. Since <strong>68 x</strong> represents the amount earned, in dollars, from consulting for <strong>x</strong> hours, it follows that the interior designer earned <strong>dollar sign 68</strong> per hour consulting last week.<br>Choice B is incorrect. The interior designer worked <strong>y</strong> hours, not <strong>68</strong> hours, drawing up plans last week.<br>Choice C is incorrect. The interior designer earned <strong>dollar sign 85</strong> per hour, not <strong>dollar sign 68</strong> per hour, drawing up plans last week.<br>Choice D is incorrect. The interior designer worked <strong>x</strong> hours, not <strong>68</strong> hours, consulting last week.",
    hasFigure: false,
  },
  {
    id: "789975b7",
    type: "mcq",
    questionHtml:
      "A gardener buys two kinds of fertilizer. Fertilizer A contains 60% filler materials by weight and Fertilizer B contains 40% filler materials by weight. Together, the fertilizers bought by the gardener contain a total of 240 pounds of filler materials. Which equation models this relationship, where x is the number of pounds of Fertilizer A and y is the number of pounds of Fertilizer B?",
    choices: [
      { label: "A", text: "<strong>zero . 4 x + zero . 6 y = 240</strong>" },
      { label: "B", text: "<strong>zero . 6 x + zero . 4 y = 240</strong>" },
      { label: "C", text: "<strong>40 x + 60 y = 240</strong>" },
      { label: "D", text: "<strong>60 x + 40 y = 240</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Since Fertilizer A contains 60% filler materials by weight, it follows that x pounds of Fertilizer A consists of 0.6x pounds of filler materials. Similarly, y pounds of Fertilizer B consists of 0.4y pounds of filler materials. When x pounds of Fertilizer A and y pounds of Fertilizer B are combined, the result is 240 pounds of filler materials. Therefore, the total amount, in pounds, of filler materials in a mixture of x pounds of Fertilizer A and y pounds of Fertilizer B can be expressed as <strong>0 . 6 x + 0 . 4 y = 240</strong>.Choice A is incorrect. This choice transposes the percentages of filler materials for Fertilizer A and Fertilizer B. Fertilizer A consists of 0.6x pounds of filler materials and Fertilizer B consists of 0.4y pounds of filler materials. Therefore, <strong>0 . 6 x + 0 . 4 y</strong> is equal to 240, not <strong>0 . 4 x + 0 . 6 y</strong>. Choice C is incorrect. This choice transposes the percentages of filler materials for Fertilizer A and Fertilizer B and incorrectly represents how to take the percentage of a value mathematically. Choice D is incorrect. This choice incorrectly represents how to take the percentage of a value mathematically. Fertilizer A consists of 0.6x pounds of filler materials, not 60x pounds of filler materials, and Fertilizer B consists of 0.4y pounds of filler materials, not 40y pounds of filler materials.",
    hasFigure: false,
  },
  {
    id: "8368afd1",
    type: "mcq",
    questionHtml:
      "The line slants sharply down from left to right.<br>The line passes through the following points:<br><br>(0 comma 18)<br>(4 comma 10)<br>(7 comma 4)<br>(9 comma 0)<br><br>The graph shows the possible combinations of the number of pounds of tangerines and lemons that could be purchased for <strong>dollar sign 18</strong> at a certain store. If Melvin purchased lemons and <strong>4</strong> pounds of tangerines for a total of <strong>dollar sign 18</strong>, how many pounds of lemons did he purchase?",
    choices: [
      { label: "A", text: "<strong>7</strong>" },
      { label: "B", text: "<strong>10</strong>" },
      { label: "C", text: "<strong>14</strong>" },
      { label: "D", text: "<strong>16</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It's given that the graph shows the possible combinations of the number of pounds of tangerines, <strong>x</strong>, and the number of pounds of lemons, <strong>y</strong>, that could be purchased for <strong>dollar sign 18</strong> at a certain store. If Melvin purchased lemons and <strong>4</strong> pounds of tangerines for a total of <strong>dollar sign 18</strong>, the number of pounds of lemons he purchased is represented by the y-coordinate of the point on the graph where <strong>x = 4</strong>. For the graph shown, when <strong>x = 4</strong>, <strong>y = 10</strong>. Therefore, if Melvin purchased lemons and <strong>4</strong> pounds of tangerines for a total of <strong>dollar sign 18</strong>, then he purchased <strong>10</strong> pounds of lemons.<br>Choice A is incorrect. This is the number of pounds of tangerines Melvin purchased if he purchased tangerines and <strong>4</strong> pounds of lemons for a total of <strong>dollar sign 18</strong>.<br>Choice C is incorrect. This is the number of pounds of lemons Melvin purchased if he purchased lemons and <strong>2</strong> pounds of tangerines for a total of <strong>dollar sign 18</strong>.<br>Choice D is incorrect. This is the number of pounds of lemons Melvin purchased if he purchased lemons and <strong>1</strong> pound of tangerines for a total of <strong>dollar sign 18</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/8368afd1_svg1.svg",
  },
  {
    id: "83f2c3bf",
    type: "mcq",
    questionHtml:
      "<strong>y = x + 4</strong><br>Which table gives three values of <strong>x</strong> and their corresponding values of <strong>y</strong> for the given equation?",
    choices: [
      {
        label: "A",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>0</strong><br><strong>4</strong><br><br><strong>1</strong><br><strong>5</strong><br><br><strong>2</strong><br><strong>6</strong>",
      },
      {
        label: "B",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>0</strong><br><strong>6</strong><br><br><strong>1</strong><br><strong>5</strong><br><br><strong>2</strong><br><strong>4</strong>",
      },
      {
        label: "C",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>0</strong><br><strong>2</strong><br><br><strong>1</strong><br><strong>1</strong><br><br><strong>2</strong><br><strong>0</strong>",
      },
      {
        label: "D",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>0</strong><br><strong>0</strong><br><br><strong>1</strong><br><strong>1</strong><br><br><strong>2</strong><br><strong>2</strong>",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Substituting <strong>0</strong> for <strong>x</strong> into the given equation yields <strong>y = 0 + 4</strong>, or <strong>y = 4</strong>. Therefore, when <strong>x = 0</strong>, the corresponding value of <strong>y</strong> for the given equation is <strong>4</strong>. Substituting <strong>1</strong> for <strong>x</strong> into the given equation yields <strong>y = 1 + 4</strong>, or <strong>y = 5</strong>. Therefore, when <strong>x = 1</strong>, the corresponding value of <strong>y</strong> for the given equation is <strong>5</strong>. Substituting <strong>2</strong> for <strong>x</strong> into the given equation yields <strong>y = 2 + 4</strong>, or <strong>y = 6</strong>. Therefore, when <strong>x = 2</strong>, the corresponding value of <strong>y</strong> for the given equation is <strong>6</strong>. Of the choices given, only the table in choice A gives these three values of <strong>x</strong> and their corresponding values of <strong>y</strong> for the given equation.<br>Choice B is incorrect. This table gives three values of <strong>x</strong> and their corresponding values of <strong>y</strong> for the equation <strong>y = −x + 6</strong>.<br>Choice C is incorrect. This table gives three values of <strong>x</strong> and their corresponding values of <strong>y</strong> for the equation <strong>y = −x + 2</strong>.<br>Choice D is incorrect. This table gives three values of <strong>x</strong> and their corresponding values of <strong>y</strong> for the equation <strong>y = x</strong>.",
    hasFigure: false,
  },
  {
    id: "87322577",
    type: "mcq",
    questionHtml:
      "The equation above relates the number of minutes, x, Maria spends running each day and the number of minutes, y, she spends biking each day. In the equation, what does the number 75 represent?",
    choices: [
      { label: "A", text: "The number of minutes spent running each day" },
      { label: "B", text: "The number of minutes spent biking each day" },
      {
        label: "C",
        text: "The total number of minutes spent running and biking each day",
      },
      {
        label: "D",
        text: "The number of minutes spent biking for each minute spent running",
      },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Maria spends x minutes running each day and y minutes biking each day. Therefore, <strong>x + y</strong> represents the total number of minutes Maria spent running and biking each day. Because <strong>x + y = 75</strong>, it follows that 75 is the total number of minutes that Maria spent running and biking each day.Choices A and B are incorrect. The number of minutes Maria spent running each day is represented by x and need not be 75. Similarly, the number of minutes that Maria spends biking each day is represented by y and need not be 75. The number of minutes Maria spends running each day and biking each day may vary; however, the total number of minutes she spends each day on these activities is constant and equal to 75. Choice D is incorrect. The number of minutes Maria spent biking for each minute spent running cannot be determined from the information provided.",
    hasFigure: false,
  },
  {
    id: "8a1544f1",
    type: "mcq",
    questionHtml:
      "What is the equation of the line shown in the xy-plane above?",
    choices: [
      { label: "A", text: "<strong>y = 3 x − 3</strong>" },
      { label: "B", text: "<strong>y = −3 x + 3</strong>" },
      { label: "C", text: "<strong>y = one-third x − 3</strong>" },
      { label: "D", text: "<strong>y = −one-third x + 3</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Any line in the xy-plane can be defined by an equation in the form y = mx + b, where m is the slope of the line and b is the y-coordinate of the y-intercept of the line. From the graph, the y-intercept of the line is (0, 3). Therefore, b = 3. The slope of the line is the change in the value of y divided by the change in the value of x for any two points on the line. The line shown passes through (0, 3) and (1, 0), so <strong>m = the fraction with numerator 3 − zero, and denominator zero − 1</strong>, or m = –3. Therefore, the equation of the line is y = –3x + 3.Choices A and C are incorrect because the equations given in these choices represent a line with a positive slope. However, the line shown has a negative slope. Choice D is incorrect because the equation given in this choice represents a line with slope of <strong>−one third</strong>. However, the line shown has a slope of  –3.",
    hasFigure: true,
    figureUrl: "/practice-images/8a1544f1_img1.png",
  },
  {
    id: "8adf1335",
    type: "mcq",
    questionHtml:
      "A city’s total expense budget for one year was x million dollars. The city budgeted y million dollars for departmental expenses and 201 million dollars for all other expenses. Which of the following represents the relationship between x and y in this context?",
    choices: [
      { label: "A", text: "<strong>x + y = 201</strong>" },
      { label: "B", text: "<strong>x − y = 201</strong>" },
      { label: "C", text: "<strong>2 x − y = 201</strong>" },
      { label: "D", text: "<strong>y − x = 201</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Of the city’s total expense budget for one year, the city budgeted y million dollars for departmental expenses and 201 million dollars for all other expenses. This means that the expression <strong>y + 201</strong> represents the total expense budget, in millions of dollars, for one year. It’s given that the total expense budget for one year is x million dollars. It follows then that the expression <strong>y + 201</strong> is equivalent to x, or <strong>y + 201 = x</strong>. Subtracting y from both sides of this equation yields <strong>201 = x − y</strong>. By the symmetric property of equality, this is the same as <strong>x − y = 201</strong>.Choices A and C are incorrect. Because it’s given that the total expense budget for one year, x million dollars, is comprised of the departmental expenses, y million dollars, and all other expenses, 201 million dollars, the expressions <strong>x + y</strong> and <strong>2 x − y</strong> both must be equivalent to a value greater than 201 million dollars. Therefore, the equations <strong>x + y = 201</strong>and <strong>2 x − y = 201</strong>aren’t true. Choice D is incorrect. The value of x must be greater than the value of y. Therefore, <strong>y − x = 201</strong> can’t represent this relationship.",
    hasFigure: false,
  },
  {
    id: "8b2a2a63",
    type: "spr",
    questionHtml:
      "The y-intercept of the graph of <strong>y = − 6 x − 32</strong> in the xy-plane is <strong>(0, y)</strong>. What is the value of <strong>y</strong>?",
    choices: [],
    correctAnswer: "-32",
    explanation:
      "The correct answer is <strong>−32</strong>. It’s given that the y-intercept of the graph of <strong>y = − 6 x − 32</strong> is <strong>(0, y)</strong>. Substituting <strong>0</strong> for <strong>x</strong> in this equation yields <strong>y = − 6 (0) − 32</strong>, or <strong>y = −32</strong>. Therefore, the value of <strong>y</strong> that corresponds to the y-intercept of the graph of <strong>y = − 6 x − 32</strong> in the xy-plane is <strong>−32</strong>.",
    hasFigure: false,
  },
  {
    id: "8c98c834",
    type: "mcq",
    questionHtml:
      "The equation <strong>y = 0 . 1 x</strong> models the relationship between the number of different pieces of music a certain pianist practices, y, during an x-minute practice session. How many pieces did the pianist practice if the session lasted 30 minutes?",
    choices: [
      { label: "A", text: "1" },
      { label: "B", text: "3" },
      { label: "C", text: "10" },
      { label: "D", text: "30" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It’s given that the equation <strong>y = 0 . 1 x</strong> models the relationship between the number of different pieces of music a certain pianist practices, y, and the number of minutes in a practice session, x. Since it’s given that the session lasted 30 minutes, the number of pieces the pianist practiced can be found by substituting 30 for x in the given equation, which yields <strong>y = 0 . 1 · 30</strong>, or <strong>y = 3</strong>.Choices A and C are incorrect and may result from misinterpreting the values in the equation. Choice D is incorrect. This is the given value of x, not the value of y.",
    hasFigure: false,
  },
  {
    id: "b23bba4c",
    type: "mcq",
    questionHtml:
      "A shipping company charged a customer $25 to ship some small boxes and some large boxes. The equation above represents the relationship between a, the number of small boxes, and b, the number of large boxes, the customer had shipped. If the customer had 3 small boxes shipped, how many large boxes were shipped?",
    choices: [
      { label: "A", text: "3" },
      { label: "B", text: "4" },
      { label: "C", text: "5" },
      { label: "D", text: "6" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It’s given that a represents the number of small boxes and b represents the number of large boxes the customer had shipped. If the customer had 3 small boxes shipped, then <strong>a = 3</strong>. Substituting 3 for a in the equation <strong>3 a + 4 b = 25</strong> yields <strong>3 · 3 + 4 b = 25</strong> or <strong>9 + 4 b = 25</strong>. Subtracting 9 from both sides of the equation yields <strong>4 b = 16</strong>. Dividing both sides of this equation by 4 yields <strong>b = 4</strong>. Therefore, the customer had 4 large boxes shipped.Choices A, C, and D are incorrect. If the number of large boxes shipped is 3, then <strong>b = 3</strong>. Substituting 3 for b in the given equation yields <strong>3 a + 4 · 3 = 25</strong> or <strong>3 a + 12 = 25</strong>. Subtracting 12 from both sides of the equation and then dividing by 3 yields <strong>a = thirteen thirds</strong>. However, it’s given that the number of small boxes shipped, a, is 3, not <strong>thirteen thirds</strong>, so b cannot equal 3. Similarly, if <strong>b = 5</strong> or <strong>b = 6</strong>, then <strong>a = five thirds</strong> or <strong>a = one third</strong>, respectively, which is also not true.",
    hasFigure: false,
  },
  {
    id: "b2845d88",
    type: "mcq",
    questionHtml:
      "Which of the following is an equation of the graph shown in the xy-plane above?",
    choices: [
      { label: "A", text: "<strong>y = −one fourth x − 1</strong>" },
      { label: "B", text: "<strong>y = −x − 4</strong>" },
      { label: "C", text: "<strong>y = −x − one fourth</strong>" },
      { label: "D", text: "<strong>y = −4 x − 1</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The slope of the line can be found by choosing any two points on the line, such as (4, –2) and (0, –1). Subtracting the y-values results in –2 – (–1) = –1, the change in y. Subtracting the x-values results in 4 – 0 = 4, the change in x. Dividing the change in y by the change in x yields <strong>−1 ÷ 4 = −one fourth</strong>, the slope. The line intersects the y-axis at (0, –1), so –1 is the y-coordinate of the y-intercept. This information can be expressed in slope-intercept form as the equation <strong>y = −one fourth, x − 1</strong>.Choice B is incorrect and may result from incorrectly calculating the slope and then misidentifying the slope as the y-intercept. Choice C is incorrect and may result from misidentifying the slope as the y-intercept. Choice D is incorrect and may result from incorrectly calculating the slope.",
    hasFigure: true,
    figureUrl: "/practice-images/b2845d88_img1.png",
  },
  {
    id: "b2de69bd",
    type: "mcq",
    questionHtml:
      "The table above shows some pairs of x values and y values. Which of the following equations could represent the relationship between x and y ?",
    choices: [
      { label: "A", text: "<strong>y = 2 x + 3</strong>" },
      { label: "B", text: "<strong>y = 3 x − 2</strong>" },
      { label: "C", text: "<strong>y = 4 x − one</strong>" },
      { label: "D", text: "<strong>y = 5 x</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Each of the choices is a linear equation in the form y = mx + b, where m and b are constants. In this equation, m represents the change in y for each increase in x by 1. From the table, it can be determined that the value of y increases by 2 for each increase in x by 1. In other words, for the  pairs of x and y in the given table, m = 2. The value of b can be found by substituting the values of x and y from any row of the table and substituting the value of m into the equation y = mx + b and then solving for b. For example, using x = 1, y = 5, and m = 2 yields 5 = 2(1) + b. Solving for b yields b = 3. Therefore, the equation y = 2x + 3 could represent the relationship between x and y in the given table.Alternatively, if an equation represents the relationship between x and y, then when each pair of x and y from the table is substituted into the equation, the result will be a true statement. Of the equations given, the equation y = 2x + 3 in choice A is the only equation that results in a true statement when each of the pairs of x and y are substituted into the equation.<br>Choices B, C, and D are incorrect because when at least one pair of x and y from the table is substituted into the equations given in these choices, the result is a false statement. For example, when the pair x = 4 and y = 11 is substituted into the equation in choice B, the result is 11 = 3(4) – 2, or 11 = 10, which is false.",
    hasFigure: false,
  },
  {
    id: "b450ab03",
    type: "mcq",
    questionHtml:
      "An employee at a restaurant prepares sandwiches and salads. It takes the employee <strong>1.5</strong> minutes to prepare a sandwich and <strong>1.9</strong> minutes to prepare a salad. The employee spends a total of <strong>46.1</strong> minutes preparing <strong>x</strong> sandwiches and <strong>y</strong> salads. Which equation represents this situation?",
    choices: [
      { label: "A", text: "<strong>1.9 x + 1.5 y = 46.1</strong>" },
      { label: "B", text: "<strong>1.5 x + 1.9 y = 46.1</strong>" },
      { label: "C", text: "<strong>x + y = 46.1</strong>" },
      { label: "D", text: "<strong>30.7 x + 24.3 y = 46.1</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It’s given that the employee takes <strong>1.5</strong> minutes to prepare a sandwich. Multiplying <strong>1.5</strong> by the number of sandwiches, <strong>x</strong>, yields <strong>1.5 x</strong>, the amount of time the employee spends preparing <strong>x</strong> sandwiches. It’s also given that the employee takes <strong>1.9</strong> minutes to prepare a salad. Multiplying <strong>1.9</strong> by the number of salads, <strong>y</strong>, yields <strong>1.9 y</strong>, the amount of time the employee spends preparing <strong>y</strong> salads. It follows that the total amount of time, in minutes, the employee spends preparing <strong>x</strong> sandwiches and <strong>y</strong> salads is <strong>1.5 x + 1.9 y</strong>. It's given that the employee spends a total of <strong>46.1</strong> minutes preparing <strong>x</strong> sandwiches and <strong>y</strong> salads. Thus, the equation <strong>1.5 x + 1.9 y = 46.1</strong> represents this situation.<br>Choice A is incorrect. This equation represents a situation where it takes the employee <strong>1.9</strong> minutes, rather than <strong>1.5</strong> minutes, to prepare a sandwich and <strong>1.5</strong> minutes, rather than <strong>1.9</strong> minutes, to prepare a salad.<br>Choice C is incorrect. This equation represents a situation where it takes the employee <strong>1</strong> minute, rather than <strong>1.5</strong> minutes, to prepare a sandwich and <strong>1</strong> minute, rather than <strong>1.9</strong> minutes, to prepare a salad.<br>Choice D is incorrect. This equation represents a situation where it takes the employee <strong>30.7</strong> minutes, rather than <strong>1.5</strong> minutes, to prepare a sandwich and <strong>24.3</strong> minutes, rather than <strong>1.9</strong> minutes, to prepare a salad.",
    hasFigure: false,
  },
  {
    id: "b9839f9e",
    type: "mcq",
    questionHtml:
      "In the equation above, F represents the total amount of money, in dollars, a food truck charges for x drinks and y salads. The price, in dollars, of each drink is the same, and the price, in dollars, of each salad is the same. Which of the following is the best interpretation for the number 7.00 in this context?",
    choices: [
      { label: "A", text: "The price, in dollars, of one drink" },
      { label: "B", text: "The price, in dollars, of one salad" },
      { label: "C", text: "The number of drinks bought during the day" },
      { label: "D", text: "The number of salads bought during the day" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It’s given that <strong>2 . 5 0 x + 7 . 0 0 y</strong> is equal to the total amount of money, in dollars, a food truck charges for x drinks and y salads. Since each salad has the same price, it follows that the total charge for y salads is <strong>7 . 0 0 y</strong> dollars. When <strong>y = 1</strong>, the value of the expression <strong>7 . 0 0 y</strong> is <strong>7 . 0 0 · 1</strong>, or 7.00. Therefore, the price for one salad is 7.00 dollars.Choice A is incorrect. Since each drink has the same price, it follows that the total charge for x drinks is <strong>2 . 5 0 x</strong> dollars. Therefore, the price, in dollars, for one drink is 2.50, not 7.00. Choices C and D are incorrect. In the given equation, F represents the total charge, in dollars, when x drinks and y salads are bought at the food truck. No information is provided about the number of drinks or the number of salads that are bought during the day. Therefore, 7.00 doesn’t represent either of these quantities.",
    hasFigure: false,
  },
  {
    id: "ba79f10f",
    type: "mcq",
    questionHtml:
      "<strong>x</strong><br><strong>y</strong><br><br><strong>0</strong><br><strong>18</strong><br><br><strong>1</strong><br><strong>13</strong><br><br><strong>2</strong><br><strong>8</strong><br><br>The table shows three values of <strong>x</strong> and their corresponding values of <strong>y</strong>. There is a linear relationship between <strong>x</strong> and <strong>y</strong>. Which of the following equations represents this relationship?",
    choices: [
      { label: "A", text: "<strong>y = 18 x + 13</strong>" },
      { label: "B", text: "<strong>y = 18 x + 18</strong>" },
      { label: "C", text: "<strong>y = − 5 x + 13</strong>" },
      { label: "D", text: "<strong>y = − 5 x + 18</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. A linear relationship can be represented by an equation of the form <strong>y = m x + b</strong>, where <strong>m</strong> and <strong>b</strong> are constants. It’s given in the table that when <strong>x = 0</strong>, <strong>y = 18</strong>. Substituting <strong>0</strong> for <strong>x</strong> and <strong>18</strong> for <strong>y</strong> in <strong>y = m x + b</strong> yields <strong>18 = m (0) + b</strong>, or <strong>18 = b</strong>. Substituting <strong>18</strong> for <strong>b</strong> in the equation <strong>y = m x + b</strong> yields <strong>y = m x + 18</strong>. It’s also given in the table that when <strong>x = 1</strong>, <strong>y = 13</strong>. Substituting <strong>1</strong> for <strong>x</strong> and <strong>13</strong> for <strong>y</strong> in the equation <strong>y = m x + 18</strong> yields <strong>13 = m (1) + 18</strong>, or <strong>13 = m + 18</strong>. Subtracting <strong>18</strong> from both sides of this equation yields <strong>−5 = m</strong>. Therefore, the equation <strong>y = − 5 x + 18</strong> represents the relationship between <strong>x</strong> and <strong>y</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "c5479c1a",
    type: "mcq",
    questionHtml:
      "A shipment consists of <strong>5</strong>-pound boxes and <strong>10</strong>-pound boxes with a total weight of <strong>220</strong> pounds. There are <strong>13</strong> <strong>10</strong>-pound boxes in the shipment. How many <strong>5</strong>-pound boxes are in the shipment?",
    choices: [
      { label: "A", text: "<strong>5</strong>" },
      { label: "B", text: "<strong>10</strong>" },
      { label: "C", text: "<strong>13</strong>" },
      { label: "D", text: "<strong>18</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It's given that the shipment consists of <strong>5</strong>-pound boxes and <strong>10</strong>-pound boxes with a total weight of <strong>220</strong> pounds. Let <strong>x</strong> represent the number of <strong>5</strong>-pound boxes and <strong>y</strong> represent the number of <strong>10</strong>-pound boxes in the shipment. Therefore, the equation <strong>5 x + 10 y = 220</strong> represents this situation. It's given that there are <strong>13</strong> <strong>10</strong>-pound boxes in the shipment. Substituting <strong>13</strong> for <strong>y</strong> in the equation <strong>5 x + 10 y = 220</strong> yields <strong>5 x + 10 (13) = 220</strong>, or <strong>5 x + 130 = 220</strong>. Subtracting <strong>130</strong> from both sides of this equation yields <strong>5 x = 90</strong>. Dividing both sides of this equation by <strong>5</strong> yields <strong>18</strong>. Thus, there are <strong>18</strong> <strong>5</strong>-pound boxes in the shipment.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect. This is the number of <strong>10</strong>-pound boxes in the shipment.",
    hasFigure: false,
  },
  {
    id: "c6b151d4",
    type: "mcq",
    questionHtml:
      "A total of <strong>364</strong> paper straws of equal length were used to construct two types of polygons: triangles and rectangles. The triangles and rectangles were constructed so that no two polygons had a common side. The equation <strong>3 x + 4 y = 364</strong> represents this situation, where <strong>x</strong> is the number of triangles constructed and <strong>y</strong> is the number of rectangles constructed. What is the best interpretation of <strong>(x, y) = (24, 73)</strong> in this context?",
    choices: [
      {
        label: "A",
        text: "If <strong>24</strong> triangles were constructed, then <strong>73</strong> rectangles were constructed.",
      },
      {
        label: "B",
        text: "If <strong>24</strong> triangles were constructed, then <strong>73</strong> paper straws were used.",
      },
      {
        label: "C",
        text: "If <strong>73</strong> triangles were constructed, then <strong>24</strong> rectangles were constructed.",
      },
      {
        label: "D",
        text: "If <strong>73</strong> triangles were constructed, then <strong>24</strong> paper straws were used.",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It's given that <strong>364</strong> paper straws of equal length were used to construct triangles and rectangles, where no two polygons had a common side. It's also given that the equation <strong>3 x + 4 y = 364</strong> represents this situation, where <strong>x</strong> is the number of triangles constructed and <strong>y</strong> is the number of rectangles constructed. The equation <strong>(x, y) = (24, 73)</strong> means that if <strong>x = 24</strong>, then <strong>y = 73</strong>. Substituting <strong>24</strong> for <strong>x</strong> and <strong>73</strong> for <strong>y</strong> in <strong>3 x + 4 y = 364</strong> yields <strong>3 (24) + 4 (73) = 364</strong>, or <strong>364 = 364</strong>, which is true. Therefore, in this context, the equation <strong>(x, y) = (24, 73)</strong> means that if <strong>24</strong> triangles were constructed, then <strong>73</strong> rectangles were constructed. <br>Choice B is incorrect and may result from conceptual errors.<br>Choice C is incorrect and may result from conceptual errors.<br>Choice D is incorrect and may result from conceptual errors.",
    hasFigure: false,
  },
  {
    id: "c8e0f511",
    type: "mcq",
    questionHtml:
      "For a camping trip a group bought <strong>x</strong> one-liter bottles of water and <strong>y</strong> three-liter bottles of water, for a total of <strong>240</strong> liters of water. Which equation represents this situation?",
    choices: [
      { label: "A", text: "<strong>x + 3 y = 240</strong>" },
      { label: "B", text: "<strong>x + y = 240</strong>" },
      { label: "C", text: "<strong>3 x + 3 y = 240</strong>" },
      { label: "D", text: "<strong>3 x + y = 240</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It's given that for a camping trip a group bought <strong>x</strong> one-liter bottles of water and <strong>y</strong> three-liter bottles of water. Since the group bought <strong>x</strong> one-liter bottles of water, the total number of liters bought from <strong>x</strong> one-liter bottles of water is represented as <strong>1 x</strong>, or <strong>x</strong>. Since the group bought <strong>y</strong> three-liter bottles of water, the total number of liters bought from <strong>y</strong> three-liter bottles of water is represented as <strong>3 y</strong>. It's given that the group bought a total of <strong>240</strong> liters; thus, the equation <strong>x + 3 y = 240</strong> represents this situation.<br>Choice B is incorrect and may result from conceptual errors.<br>Choice C is incorrect and may result from conceptual errors.<br>Choice D is incorrect. This equation represents a situation where the group bought <strong>x</strong> three-liter bottles of water and <strong>y</strong> one-liter bottles of water, for a total of <strong>240</strong> liters of water.",
    hasFigure: false,
  },
  {
    id: "db0107df",
    type: "spr",
    questionHtml:
      "The y-intercept of the graph of <strong>12 x + 2 y = 18</strong> in the xy-plane is <strong>(0, y)</strong>. What is the value of <strong>y</strong>?",
    choices: [],
    correctAnswer: "9",
    explanation:
      "The correct answer is <strong>9</strong>. It's given that the y-intercept of the graph of <strong>12 x + 2 y = 18</strong> in the xy-plane is <strong>(0, y)</strong>. Substituting <strong>0</strong> for <strong>x</strong> in the equation <strong>12 x + 2 y = 18</strong> yields <strong>12 (0) + 2 y = 18</strong>, or <strong>2 y = 18</strong>. Dividing both sides of this equation by <strong>2</strong> yields <strong>y = 9</strong>. Therefore, the value of <strong>y</strong> is <strong>9</strong>.",
    hasFigure: false,
  },
  {
    id: "dd797fe2",
    type: "mcq",
    questionHtml:
      "Mario purchased 4 binders that cost x dollars each and 3 notebooks that cost y dollars each. If the given equation represents this situation, which of the following is the best interpretation of 24 in this context?",
    choices: [
      {
        label: "A",
        text: "The total cost, in dollars, for all binders purchased",
      },
      {
        label: "B",
        text: "The total cost, in dollars, for all notebooks purchased",
      },
      {
        label: "C",
        text: "The total cost, in dollars, for all binders and notebooks purchased",
      },
      {
        label: "D",
        text: "The difference in the total cost, in dollars, between the number of binders and notebooks purchased",
      },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Since Mario purchased 4 binders that cost x dollars each, the expression <strong>4 x</strong> represents the total cost, in dollars, of the 4 binders he purchased. Since Mario purchased 3 notebooks that cost y dollars each, the expression <strong>3 y</strong> represents the total cost, in dollars, of the 3 notebooks he purchased. Therefore, the expression <strong>4 x + 3 y</strong> represents the total cost, in dollars, for all binders and notebooks he purchased. In the given equation, the expression <strong>4 x + 3 y</strong> is equal to 24. Therefore, it follows that 24 is the total cost, in dollars, for all binders and notebooks purchased.Choice A is incorrect. This is represented by the expression <strong>4 x</strong> in the given equation. Choice B is incorrect. This is represented by the expression <strong>3 y</strong> in the given equation. Choice D is incorrect. This is represented by the expression <strong>|, 4 x − 3 y, |</strong>.",
    hasFigure: false,
  },
  {
    id: "dfa45424",
    type: "mcq",
    questionHtml:
      "Tony spends $80 per month on public transportation. A 10-ride pass costs $12.50, and a single-ride pass costs $1.50. If g represents the number of 10-ride passes Tony buys in a month and t represents the number of single-ride passes Tony buys in a month, which of the following equations best represents the relationship between g and t ?",
    choices: [
      { label: "A", text: "<strong>g + t = 80</strong>" },
      { label: "B", text: "<strong>g + t = 1 . 5 0 + 12 . 5 0</strong>" },
      { label: "C", text: "<strong>1 . 5 0 g + 12 . 5 0 t = 80</strong>" },
      { label: "D", text: "<strong>12 . 5 0 g + 1 . 5 0 t = 80</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Since a 10-ride pass costs $12.50 and g is the number of 10-ride passes Tony buys in a month, the expression <strong>12 . 5 0 g</strong> represents the amount Tony spends on 10-ride passes in a month. Since a single-ride pass costs $1.50 and t is the number of single-ride passes Tony buys in a month, the expression <strong>1 . 5 0 t</strong> represents the amount Tony spends on single-ride passes in a month. Therefore, the sum <strong>12 . 5 0 g + 1 . 5 0 t</strong> represents the amount he spends on the two types of passes in a month. Since Tony spends a total of $80 on passes in a month, this expression can be set equal to 80, producing <strong>12 . 5 0 g + 1 . 5 0 t = 80</strong>.Choices A and B are incorrect. The expression <strong>g + t</strong> represents the total number of the two types of passes Tony buys in a month, not the amount Tony spends, which is equal to 80, nor the cost of one of each pass, which is equal to <strong>1 . 5 0 + 12 . 5 0</strong>. Choice C is incorrect and may result from reversing the cost for each type of pass Tony buys in a month.",
    hasFigure: false,
  },
  {
    id: "e7343559",
    type: "mcq",
    questionHtml:
      "<strong>y = − 4 x + 40</strong><br>Which table gives three values of <strong>x</strong> and their corresponding values of <strong>y</strong> for the given equation?",
    choices: [
      {
        label: "A",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>0</strong><br><strong>0</strong><br><br><strong>1</strong><br><strong>−4</strong><br><br><strong>2</strong><br><strong>−8</strong>",
      },
      {
        label: "B",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>0</strong><br><strong>40</strong><br><br><strong>1</strong><br><strong>44</strong><br><br><strong>2</strong><br><strong>48</strong>",
      },
      {
        label: "C",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>0</strong><br><strong>40</strong><br><br><strong>1</strong><br><strong>36</strong><br><br><strong>2</strong><br><strong>32</strong>",
      },
      {
        label: "D",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>0</strong><br><strong>0</strong><br><br><strong>1</strong><br><strong>4</strong><br><br><strong>2</strong><br><strong>8</strong>",
      },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Each of the given choices gives three values of <strong>x</strong>: <strong>0</strong>, <strong>1</strong>, and <strong>2</strong>. Substituting <strong>0</strong> for <strong>x</strong> in the given equation yields <strong>y = − 4 (0) + 40</strong>, or <strong>y = 40</strong>. Therefore, when <strong>x = 0</strong>, the corresponding value of <strong>y</strong> for the given equation is <strong>40</strong>. Substituting <strong>1</strong> for <strong>x</strong> in the given equation yields <strong>y = − 4 (1) + 40</strong>, or <strong>y = 36</strong>. Therefore, when <strong>x = 1</strong>, the corresponding value of <strong>y</strong> for the given equation is <strong>36</strong>. Substituting <strong>2</strong> for <strong>x</strong> in the given equation yields <strong>y = − 4 (2) + 40</strong>, or <strong>y = 32</strong>. Therefore, when <strong>x = 2</strong>, the corresponding value of <strong>y</strong> for the given equation is <strong>32</strong>. Choice C gives three values of <strong>x</strong>, <strong>0</strong>, <strong>1</strong>, and <strong>2</strong>, and their corresponding values of <strong>y</strong>, <strong>40</strong>, <strong>36</strong>, and <strong>32</strong>, respectively, for the given equation.<br>Choice A is incorrect. This table gives three values of <strong>x</strong> and their corresponding values of <strong>y</strong> for the equation <strong>y = − 4 x</strong>.<br>Choice B is incorrect. This table gives three values of <strong>x</strong> and their corresponding values of <strong>y</strong> for the equation <strong>y = 4 x + 40</strong>.<br>Choice D is incorrect. This table gives three values of <strong>x</strong> and their corresponding values of <strong>y</strong> for the equation <strong>y = 4 x</strong>.",
    hasFigure: false,
  },
  {
    id: "ebf8d2b7",
    type: "mcq",
    questionHtml:
      "A machine makes large boxes or small boxes, one at a time, for a total of <strong>700</strong> minutes each day. It takes the machine <strong>10</strong> minutes to make a large box or <strong>5</strong> minutes to make a small box. Which equation represents the possible number of large boxes, <strong>x</strong>, and small boxes, <strong>y</strong>, the machine can make each day?",
    choices: [
      { label: "A", text: "<strong>5 x + 10 y = 700</strong>" },
      { label: "B", text: "<strong>10 x + 5 y = 700</strong>" },
      { label: "C", text: "<strong>(x + y) (10 + 5) = 700</strong>" },
      { label: "D", text: "<strong>(10 + x) (5 + y) = 700</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It’s given that it takes the machine <strong>10</strong> minutes to make a large box. It's also given that <strong>x</strong> represents the possible number of large boxes the machine can make each day. Multiplying <strong>10</strong> by <strong>x</strong> gives <strong>10 x</strong>, which represents the amount of time spent making large boxes. It’s given that it takes the machine <strong>5</strong> minutes to make a small box. It's also given that <strong>y</strong> represents the possible number of small boxes the machine can make each day. Multiplying <strong>5</strong> by <strong>y</strong> gives <strong>5 y</strong>, which represents the amount of time spent making small boxes. Combining the amount of time spent making <strong>x</strong> large boxes and <strong>y</strong> small boxes yields <strong>10 x + 5 y</strong>. It’s given that the machine makes boxes for a total of <strong>700</strong> minutes each day. Therefore <strong>10 x + 5 y = 700</strong> represents the possible number of large boxes, <strong>x</strong>, and small boxes, <strong>y</strong>, the machine can make each day.<br>Choice A is incorrect and may result from associating the time of <strong>10</strong> minutes with small, rather than large, boxes and the time of <strong>5</strong> minutes with large, rather than small, boxes.<br>Choice C is incorrect and may result from conceptual errors.<br>Choice D is incorrect and may result from conceptual errors.",
    hasFigure: false,
  },
  {
    id: "ee846db7",
    type: "mcq",
    questionHtml:
      "A store sells two different-sized containers of a certain Greek yogurt. The store’s sales of this Greek yogurt totaled <strong>1, 277.94</strong> dollars last month. The equation <strong>5.48 x + 7.30 y = 1, 277.94</strong> represents this situation, where <strong>x</strong> is the number of smaller containers sold and <strong>y</strong> is the number of larger containers sold. According to the equation, which of the following represents the price, in dollars, of each smaller container?",
    choices: [
      { label: "A", text: "<strong>5.48</strong>" },
      { label: "B", text: "<strong>7.30 y</strong>" },
      { label: "C", text: "<strong>7.30</strong>" },
      { label: "D", text: "<strong>5.48 x</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It's given that the store's sales of a certain Greek yogurt totaled <strong>1, 277.94</strong> dollars last month. It's also given that the equation <strong>5.48 x + 7.30 y = 1, 277.94</strong> represents this situation, where <strong>x</strong> is the number of smaller containers sold and <strong>y</strong> is the number of larger containers sold. Since <strong>x</strong> represents the number of smaller containers of yogurt sold, the expression <strong>5.48 x</strong> represents the total sales, in dollars, from smaller containers of yogurt. This means that <strong>x</strong> smaller containers of yogurt were sold at a price of <strong>5.48</strong> dollars each. Therefore, according to the equation, <strong>5.48</strong> represents the price, in dollars, of each smaller container.<br>Choice B is incorrect. This expression represents the total sales, in dollars, from selling <strong>y</strong> larger containers of yogurt.<br>Choice C is incorrect. This value represents the price, in dollars, of each larger container of yogurt.<br>Choice D is incorrect. This expression represents the total sales, in dollars, from selling <strong>x</strong> smaller containers of yogurt.",
    hasFigure: false,
  },
  {
    id: "f40552a9",
    type: "mcq",
    questionHtml:
      "The line slants gradually up from left to right.<br>The line passes through the following points:<br><br>(0 comma 8)<br>(5 comma 9)<br>(10 comma 10)<br><br>What is the y-intercept of the line graphed?",
    choices: [
      { label: "A", text: "<strong>(0 −8)</strong>" },
      { label: "B", text: "<strong>(0 −one eighth)</strong>" },
      { label: "C", text: "<strong>(0, 0)</strong>" },
      { label: "D", text: "<strong>(0, 8)</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The y-intercept of a line graphed in the xy-plane is the point where the line intersects the y-axis. The line graphed intersects the y-axis at the point <strong>(0, 8)</strong>. Therefore, the y-intercept of the line graphed is <strong>(0, 8)</strong>.<br>Choice A is incorrect and may result from conceptual errors.<br>Choice B is incorrect and may result from conceptual errors.<br>Choice C is incorrect and may result from conceptual errors.",
    hasFigure: true,
    figureUrl: "/practice-images/f40552a9_svg1.svg",
  },
];

export const LINEAR_EQ_TWO_VAR_MEDIUM: PracticeQuestion[] = [
  {
    id: "002dba45",
    type: "spr",
    questionHtml:
      "Line <strong>k</strong> is defined by <strong>y = − (17) / (3) x + 5</strong>. Line <strong>j</strong> is perpendicular to line <strong>k</strong> in the xy-plane. What is the slope of line <strong>j</strong>?",
    choices: [],
    correctAnswer: ".1764, .1765, 3/17",
    explanation:
      "The correct answer is <strong>three seventeenths</strong>. It’s given that line <strong>j</strong> is perpendicular to line <strong>k</strong> in the xy-plane. This means that the slope of line <strong>j</strong> is the negative reciprocal of the slope of line <strong>k</strong>. The equation of line <strong>k</strong>, <strong>y = − (17) / (3) x + 5</strong>, is written in slope-intercept form <strong>y = m x + b</strong>, where <strong>m</strong> is the slope of the line and <strong>b</strong> is the y-coordinate of the y-intercept of the line. It follows that the slope of line <strong>k</strong> is <strong>−(17) / (3)</strong>. The negative reciprocal of a number is <strong>−1</strong> divided by the number. Therefore, the negative reciprocal of <strong>−(17) / (3)</strong> is <strong>Start(−1 OverOver − StartFraction 17) / (3) EndEndFraction</strong>, or <strong>three seventeenths</strong>. Thus, the slope of line <strong>j</strong> is <strong>three seventeenths</strong>. Note that 3/17, .1764, .1765, and 0.176 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "01682aa5",
    type: "mcq",
    questionHtml:
      "Line <strong>p</strong> is defined by <strong>2 y + 18 x = 9</strong>. Line <strong>r</strong> is perpendicular to line <strong>p</strong> in the xy-plane. What is the slope of line <strong>r</strong>?",
    choices: [
      { label: "A", text: "<strong>−9</strong>" },
      { label: "B", text: "<strong>−one ninth</strong>" },
      { label: "C", text: "<strong>one ninth</strong>" },
      { label: "D", text: "<strong>9</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It’s given that line <strong>r</strong> is perpendicular to line <strong>p</strong> in the xy-plane. This means that the slope of line <strong>r</strong> is the negative reciprocal of the slope of line <strong>p</strong>. If the equation for line <strong>p</strong> is rewritten in slope-intercept form <strong>y = m x + b</strong>, where <strong>m</strong> and <strong>b</strong> are constants, then <strong>m</strong> is the slope of the line and <strong>(0, b)</strong> is its y-intercept. Subtracting <strong>18 x</strong> from both sides of the equation <strong>2 y + 18 x = 9</strong> yields <strong>2 y = − 18 x + 9</strong>. Dividing both sides of this equation by <strong>2</strong> yields <strong>y = − 9 x + nine halves</strong>. It follows that the slope of line <strong>p</strong> is <strong>−9</strong>. The negative reciprocal of a number is <strong>−1</strong> divided by the number. Therefore, the negative reciprocal of <strong>−9</strong> is <strong>(−1) / (−9)</strong>, or <strong>one ninth</strong>. Thus, the slope of line <strong>r</strong> is <strong>one ninth</strong>.<br>Choice A is incorrect. This is the slope of line <strong>p</strong>, not line <strong>r</strong>.<br>Choice B is incorrect. This is the reciprocal, not the negative reciprocal, of the slope of line <strong>p</strong>.<br>Choice D is incorrect. This is the negative, not the negative reciprocal, of the slope of line <strong>p</strong>.",
    hasFigure: false,
  },
  {
    id: "038d87d7",
    type: "mcq",
    questionHtml:
      "A neighborhood consists of a <strong>2</strong>-hectare park and a <strong>35</strong>-hectare residential area. The total number of trees in the neighborhood is <strong>3, 934</strong>. The equation <strong>2 x + 35 y = 3, 934</strong> represents this situation. Which of the following is the best interpretation of x in this context?",
    choices: [
      {
        label: "A",
        text: "The average number of trees per hectare in the park",
      },
      {
        label: "B",
        text: "The average number of trees per hectare in the residential area",
      },
      { label: "C", text: "The total number of trees in the park" },
      { label: "D", text: "The total number of trees in the residential area" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It's given that a neighborhood consists of a <strong>2</strong>-hectare park and a <strong>35</strong>-hectare residential area and that the total number of trees in the neighborhood is <strong>3, 934</strong>. It's also given that the equation <strong>2 x + 35 y = 3, 934</strong> represents this situation. Since the total number of trees for a given area can be determined by taking the number of hectares times the average number of trees per hectare, this must mean that the terms <strong>2 x</strong> and <strong>35 y</strong> correspond to the number of trees in the park and in the residential area, respectively. Since <strong>2 x</strong> corresponds to the number of trees in the park, and <strong>2</strong> is the size of the park, in hectares, <strong>x</strong> must represent the average number of trees per hectare in the park.<br>Choice B is incorrect and may result from conceptual errors.<br>Choice C is incorrect and may result from conceptual errors.<br>Choice D is incorrect and may result from conceptual errors.",
    hasFigure: false,
  },
  {
    id: "265f2a53",
    type: "mcq",
    questionHtml:
      "When line <strong>n</strong> is graphed in the xy-plane, it has an x-intercept of <strong>(−4, 0)</strong> and a y-intercept of <strong>(0, (86) / (3))</strong>. What is the slope of line <strong>n</strong>?",
    choices: [
      { label: "A", text: "<strong>(3) / (344)</strong>" },
      { label: "B", text: "<strong>six forty thirds</strong>" },
      { label: "C", text: "<strong>(43) / (6)</strong>" },
      { label: "D", text: "<strong>(344) / (3)</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It's given that when line <strong>n</strong> is graphed in the xy-plane, it has an x-intercept of <strong>(−4, 0)</strong> and a y-intercept of <strong>(0, (86) / (3))</strong>. The slope, <strong>m</strong>, of a line can be found using any two points on the line, <strong>(x 1, y 1)</strong> and <strong>(x 2, y 2)</strong>, and the slope formula <strong>m = (y 2 − y 1) / (x 2 − x 1)</strong>. Substituting the points <strong>(−4, 0)</strong> and <strong>(0, (86) / (3))</strong> for <strong>(x 1, y 1)</strong> and <strong>(x 2, y 2)</strong>, respectively, in the slope formula yields <strong>m = Start(StartFraction 86) / (3) − 0 OverOver 0 − (−4) EndEndFraction</strong>, or <strong>m = (43) / (6)</strong>. Therefore, the slope of line <strong>n</strong> is <strong>(43) / (6)</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect. This is the slope of a line that has an x-intercept of <strong>((86) / (3), 0)</strong> and a y-intercept of <strong>(0 −4)</strong>.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "28c2253f",
    type: "mcq",
    questionHtml:
      "A city is planning to build a rock retaining wall, a monument, and a garden in a park. The table above shows four rock types that will be considered for use in the project. Also shown for each rock type is its weight per volume, in pounds per cubic foot (lb/ft3), and the cost per pound, in dollars. Only basalt, granite, and limestone will be used in the garden. The rocks in the garden will have a total weight of 1,000 pounds. If 330 pounds of granite is used, which of the following equations could show the relationship between the amounts, x and y, in ft3, for each of the other rock types used?",
    choices: [
      { label: "A", text: "<strong>165 x + 180 y = 670</strong>" },
      { label: "B", text: "<strong>165 x + 120 y = 1, 000</strong>" },
      { label: "C", text: "<strong>120 x + 180 y = 670</strong>" },
      { label: "D", text: "<strong>120 x + 180 y = 1, 000</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It’s given that the weight of the granite used in the garden is 330 pounds. The weight of the limestone used in the garden is a product of its weight per volume, in lb/ft3, and its volume, in ft3. Therefore, the weight of the limestone used in the garden can be represented by <strong>120 x</strong>, where x is the volume, in ft3, of the limestone used. Similarly, the weight of the basalt used in the garden can be represented by <strong>180 y</strong>, where y is the volume, in ft3, of the basalt used. It’s given that the total weight of the rocks used in the garden will be 1,000 pounds. Thus, the sum of the weights of the three rock types used is 1,000 pounds, which can be represented by the equation <strong>120 x + 180 y + 330 = 1, 000</strong>. Subtracting 330 from both sides of this equation yields <strong>120 x + 180 y = 670</strong>.Choice A is incorrect. This equation uses the weight per volume of granite instead of limestone. Choice B is incorrect. This equation uses the weight per volume of granite instead of basalt, and doesn’t take into account the 330 pounds of granite that will be used in the garden. Choice D is incorrect. This equation doesn’t take into account the 330 pounds of granite that will be used in the garden.",
    hasFigure: false,
  },
  {
    id: "2e0290c3",
    type: "mcq",
    questionHtml:
      "The line slants down from left to right.<br>The line passes through the following points:<br><br>(0 comma 40)<br>(60 comma 0)<br><br>The graph shows the relationship between the number of shares of stock from Company A, <strong>x</strong>, and the number of shares of stock from Company B, <strong>y</strong>, that Simone can purchase. Which equation could represent this relationship?",
    choices: [
      { label: "A", text: "<strong>y = 8 x + 12</strong>" },
      { label: "B", text: "<strong>8 x + 12 y = 480</strong>" },
      { label: "C", text: "<strong>y = 12 x + 8</strong>" },
      { label: "D", text: "<strong>12 x + 8 y = 480</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The graph shown is a line passing through the points <strong>(0, 40)</strong> and <strong>(60, 0)</strong>. Since the relationship between <strong>x</strong> and <strong>y</strong> is linear, if two points on the graph make a linear equation true, then the equation represents the relationship. Substituting <strong>0</strong> for <strong>x</strong> and <strong>40</strong> for <strong>y</strong> in the equation in choice B, <strong>8 x + 12 y = 480</strong>, yields <strong>8 (0) + 12 (40) = 480</strong>, or <strong>480 = 480</strong>, which is true. Substituting <strong>60</strong> for <strong>x</strong> and <strong>0</strong> for <strong>y</strong> in the equation <strong>8 x + 12 y = 480</strong> yields <strong>8 (60) + 12 (0) = 480</strong>, or <strong>480 = 480</strong>, which is true. Therefore, the equation <strong>8 x + 12 y = 480</strong> represents the relationship between <strong>x</strong> and <strong>y</strong>.<br>Choice A is incorrect. The point <strong>(0, 40)</strong> is not on the graph of this equation, since <strong>40 = 8 (0) + 12</strong>, or <strong>40 = 12</strong>, is not true.<br>Choice C is incorrect. The point <strong>(0, 40)</strong> is not on the graph of this equation, since <strong>40 = 12 (0) + 8</strong>, or <strong>40 = 8</strong>, is not true.<br>Choice D is incorrect. The point <strong>(0, 40)</strong> is not on the graph of this equation, since <strong>12 (0) + 8 (40) = 480</strong>, or <strong>320 = 480</strong>, is not true.",
    hasFigure: true,
    figureUrl: "/practice-images/2e0290c3_svg1.svg",
  },
  {
    id: "2e1a7f66",
    type: "mcq",
    questionHtml:
      "Figure A and figure B are both regular polygons. The sum of the perimeter of figure A and the perimeter of figure B is <strong>63</strong> inches. The equation <strong>3 x + 6 y = 63</strong> represents this situation, where <strong>x</strong> is the number of sides of figure A and <strong>y</strong> is the number of sides of figure B. Which statement is the best interpretation of <strong>6</strong> in this context?",
    choices: [
      {
        label: "A",
        text: "Each side of figure B has a length of <strong>6</strong> inches.",
      },
      {
        label: "B",
        text: "The number of sides of figure B is <strong>6</strong>.",
      },
      {
        label: "C",
        text: "Each side of figure A has a length of <strong>6</strong> inches.",
      },
      {
        label: "D",
        text: "The number of sides of figure A is <strong>6</strong>.",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It’s given that figure A and figure B (not shown) are both regular polygons and the sum of the perimeters of the two figures is <strong>63 inches</strong>. It’s also given that <strong>x</strong> is the number of sides of figure A and <strong>y</strong> is the number of sides of figure B, and that the equation <strong>3 x + 6 y = 63</strong> represents this situation. Thus, <strong>3 x</strong> and <strong>6 y</strong> represent the perimeters, in inches, of figure A and figure B, respectively. Since <strong>6 y</strong> represents the perimeter, in inches, of figure B and <strong>y</strong> is the number of sides of figure B, it follows that each side of figure B has a length of <strong>6 inches</strong>.<br>Choice B is incorrect. The number of sides of figure B is <strong>y</strong>, not <strong>6</strong>.<br>Choice C is incorrect. Since the perimeter, in inches, of figure A is represented by <strong>3 x</strong>, each side of figure A has a length of <strong>3 inches</strong>, not <strong>6 inches</strong>.<br>Choice D is incorrect. The number of sides of figure A is <strong>x</strong>, not <strong>6</strong>.",
    hasFigure: false,
  },
  {
    id: "431c3038",
    type: "mcq",
    questionHtml:
      "In an article about exercise, it is estimated that a 160-pound adult uses 200 calories for every 30 minutes of hiking and 150 calories for every 30 minutes of bicycling. An adult who weighs 160 pounds has completed 1 hour of bicycling. Based on the article, how many hours should the adult hike to use a total of 1,900 calories from bicycling and hiking?",
    choices: [
      { label: "A", text: "9.5" },
      { label: "B", text: "8.75" },
      { label: "C", text: "6" },
      { label: "D", text: "4" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Since a 160-pound adult uses 200 calories for every 30 minutes of hiking, then the same adult uses <strong>200 h</strong> calories after hiking for h 30-minute periods. Similarly, the same adult uses <strong>150 b</strong> calories after bicycling for b 30-minute periods. Therefore, the equation <strong>200 h + 150 b = 1, 900</strong> represents the situation where a 160-pound adult uses a total of 1,900 calories from hiking for h 30-minute periods and bicycling for b 30-minute periods. It’s given that the adult completes 1 hour, or 2 30-minute periods, of bicycling. Substituting 2 for b in the equation <strong>200 h + 150 b = 1, 900</strong> yields <strong>200 h + 150 · 2 = 1, 900</strong>, or <strong>200 h + 300 = 1, 900</strong>. Subtracting 300 from both sides of this equation yields <strong>200 h = 1, 600</strong>. Dividing both sides by 200 yields <strong>h = 8</strong>. Since h represents the number of 30-minute periods spent hiking and there are 2 30-minute periods in every hour, it follows that the adult will need to hike for <strong>the fraction eight over 2</strong>, or 4 hours to use a total of 1,900 calories from bicycling and hiking.Choice A is incorrect and may result from solving the equation <strong>200 h = 1, 900</strong>. This represents 0 30-minute periods bicycling instead of 2. Choice B is incorrect and may result from solving the equation <strong>200 h + 150 = 1, 900</strong>. This represents 1 30-minute period of bicycling instead of 2. Choice C is incorrect. This may result from determining that the number of 30-minute periods the adult should hike is 8, but then subtracting 2 from 8, rather than dividing 8 by 2, to find the number of hours the adult should hike.",
    hasFigure: false,
  },
  {
    id: "637022d2",
    type: "mcq",
    questionHtml:
      "<strong>2.5 b + 5 r = 80</strong><br>The given equation describes the relationship between the number of birds, <strong>b</strong>, and the number of reptiles, <strong>r</strong>, that can be cared for at a pet care business on a given day. If the business cares for <strong>16</strong> reptiles on a given day, how many birds can it care for on this day?",
    choices: [
      { label: "A", text: "<strong>0</strong>" },
      { label: "B", text: "<strong>5</strong>" },
      { label: "C", text: "<strong>40</strong>" },
      { label: "D", text: "<strong>80</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The number of birds can be found by calculating the value of <strong>b</strong> when <strong>r = 16</strong> in the given equation. Substituting <strong>16</strong> for <strong>r</strong> in the given equation yields <strong>2.5 b + 5 (16) = 80</strong>, or <strong>2.5 b + 80 = 80</strong>. Subtracting <strong>80</strong> from both sides of this equation yields <strong>2.5 b = 0</strong>. Dividing both sides of this equation by <strong>2.5</strong> yields <strong>b = 0</strong>. Therefore, if the business cares for <strong>16</strong> reptiles on a given day, it can care for <strong>0</strong> birds on this day.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "6f6dfe3e",
    type: "mcq",
    questionHtml:
      "<strong>x</strong><br><strong>y</strong><br><br><strong>−6</strong><br><strong>n + 184</strong><br><br><strong>−3</strong><br><strong>n + 92</strong><br><br><strong>0</strong><br><strong>n</strong><br><br>The table shows three values of <strong>x</strong> and their corresponding values of <strong>y</strong>, where <strong>n</strong> is a constant, for the linear relationship between <strong>x</strong> and <strong>y</strong>. What is the slope of the line that represents this relationship in the xy-plane?",
    choices: [
      { label: "A", text: "<strong>−(92) / (3)</strong>" },
      { label: "B", text: "<strong>−three ninety seconds</strong>" },
      { label: "C", text: "<strong>(n + 92) / (−3)</strong>" },
      { label: "D", text: "<strong>(2 n − 92) / (3)</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The slope, <strong>m</strong>, of a line in the xy-plane can be found using two points on the line, <strong>(x 1, y 1)</strong> and <strong>(x 2, y 2)</strong>, and the slope formula <strong>m = (y 2 − y 1) / (x 2 − x 1)</strong>. Based on the given table, the line representing the relationship between <strong>x</strong> and <strong>y</strong> in the xy-plane passes through the points <strong>(−6, n + 184)</strong>, <strong>(−3, n + 92)</strong>, and <strong>(0, n)</strong>, where <strong>n</strong> is a constant. Substituting two of these points, <strong>(−3, n + 92)</strong> and <strong>(0, n)</strong>, for <strong>(x 1, y 1)</strong> and <strong>(x 2, y 2)</strong>, respectively, in the slope formula yields <strong>m = (n − (n + 92)) / (0 − (−3))</strong>, which is equivalent to <strong>m = (n − n − 92) / (0 + 3)</strong>, or <strong>m = −(92) / (3)</strong>. Therefore, the slope of the line that represents this relationship in the xy-plane is <strong>−(92) / (3)</strong>.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "9c7741c6",
    type: "spr",
    questionHtml:
      "On a 210-mile trip, Cameron drove at an average speed of 60 miles per hour for the first x hours. He then completed the trip, driving at an average speed of 50 miles per hour for the remaining y hours. If <strong>x = 1</strong>, what is the value of y ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 3. It’s given that Cameron drove 60 miles per hour for x hours; therefore, the distance driven at this speed can be represented by <strong>60 x</strong>. He then drove 50 miles per hour for y hours; therefore, the distance driven at this speed can be represented by <strong>50 y</strong>. Since Cameron drove 210 total miles, the equation <strong>60 x + 50 y = 210</strong> represents this situation. If <strong>x = 1</strong>, substitution yields <strong>60 · 1 + 50 y = 210</strong>, or <strong>60 + 50 y = 210</strong>. Subtracting 60 from both sides of this equation yields <strong>50 y = 150</strong>. Dividing both sides of this equation by 50 yields <strong>y = 3.</strong>.",
    hasFigure: false,
  },
  {
    id: "9ed4c1a2",
    type: "spr",
    questionHtml:
      "What is the slope of the graph of <strong>y = one fourth (27 x + 15) + 7 x</strong> in the xy-plane?",
    choices: [],
    correctAnswer: "13.75, 55/4",
    explanation:
      "The correct answer is <strong>(55) / (4)</strong>. In the xy-plane, the graph of an equation in the form <strong>y = m x + b</strong>, where <strong>m</strong> and <strong>b</strong> are constants, has a slope of <strong>m</strong> and a y-intercept of <strong>(0, b)</strong>. Applying the distributive property to the right-hand side of the given equation yields <strong>y = (27) / (4) x + (15) / (4) + 7 x</strong>. Combining like terms yields <strong>y = (55) / (4) x + (15) / (4)</strong>. This equation is in the form <strong>y = m x + b</strong>, where <strong>m = (55) / (4)</strong> and <strong>b = (15) / (4)</strong>. It follows that the slope of the graph of <strong>y = one fourth (27 x + 15) + 7 x</strong> in the xy-plane is <strong>(55) / (4)</strong>. Note that 55/4 and 13.75 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "9f3cb472",
    type: "mcq",
    questionHtml:
      "Line <strong>t</strong> in the xy-plane has a slope of <strong>−one third</strong> and passes through the point <strong>(9, 10)</strong>. Which equation defines line <strong>t</strong>?",
    choices: [
      { label: "A", text: "<strong>y = 13 x − one third</strong>" },
      { label: "B", text: "<strong>y = 9 x + 10</strong>" },
      { label: "C", text: "<strong>y = − (x) / (3) + 10</strong>" },
      { label: "D", text: "<strong>y = − (x) / (3) + 13</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The equation that defines line <strong>t</strong> in the xy-plane can be written in slope-intercept form <strong>y = m x + b</strong>, where <strong>m</strong> is the slope of line <strong>t</strong> and <strong>(0, b)</strong> is its y-intercept. It’s given that line <strong>t</strong> has a slope of <strong>−one third</strong>. Therefore, <strong>m = −one third</strong>. Substituting <strong>−one third</strong> for <strong>m</strong> in the equation <strong>y = m x + b</strong> yields <strong>y = − one third x + b</strong>, or <strong>y = − (x) / (3) + b</strong>. It’s also given that line <strong>t</strong> passes through the point <strong>(9, 10)</strong>. Substituting <strong>9</strong> for <strong>x</strong> and <strong>10</strong> for <strong>y</strong> in the equation <strong>y = − (x) / (3) + b</strong> yields <strong>10 = −nine thirds + b</strong>, or <strong>10 = −3 + b</strong>. Adding <strong>3</strong> to both sides of this equation yields <strong>13 = b</strong>. Substituting <strong>13</strong> for <strong>b</strong> in the equation <strong>y = − (x) / (3) + b</strong> yields <strong>y = − (x) / (3) + 13</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect. This equation defines a line that has a slope of <strong>9</strong>, not <strong>−one third</strong>, and passes through the point <strong>(0, 10)</strong>, not <strong>(9, 10)</strong>.<br>Choice C is incorrect. This equation defines a line that passes through the point <strong>(0, 10)</strong>, not <strong>(9, 10)</strong>.",
    hasFigure: false,
  },
  {
    id: "c307283c",
    type: "mcq",
    questionHtml:
      "The line slants gradually down from left to right.<br>The line passes through the following points:<br><br>(negative 8 comma 0)<br>(0 comma negative 8) <br><br>What is an equation of the graph shown?",
    choices: [
      { label: "A", text: "<strong>y = − 2 x − 8</strong>" },
      { label: "B", text: "<strong>y = x − 8</strong>" },
      { label: "C", text: "<strong>y = −x − 8</strong>" },
      { label: "D", text: "<strong>y = 2 x − 8</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. An equation of a line can be written in the form <strong>y = m x + b</strong>, where <strong>m</strong> is the slope of the line and <strong>(0, b)</strong> is the y-intercept of the line. The line shown passes through the point <strong>(0 −8)</strong>, so <strong>b = −8</strong>. The line shown also passes through the point <strong>(−8, 0)</strong>. The slope, <strong>m</strong>, of a line passing through two points <strong>(x 1, y 1)</strong> and <strong>(x 2, y 2)</strong> can be calculated using the equation <strong>m = (y 2 − y 1) / (x 2 − x 1)</strong>. For the points <strong>(0 −8)</strong> and <strong>(−8, 0)</strong>, this gives <strong>m = ((−8) − 0) / (0 − (−8))</strong>, or <strong>m = −1</strong>. Substituting <strong>−8</strong> for <strong>b</strong> and <strong>−1</strong> for <strong>m</strong> in <strong>y = m x + b</strong> yields <strong>y = (−1) x + (−8)</strong>, or <strong>y = −x − 8</strong>. Therefore, an equation of the graph shown is <strong>y = −x − 8</strong>.<br>Choice A is incorrect. This is an equation of a line with a slope of <strong>−2</strong>, not <strong>−1</strong>.<br>Choice B is incorrect. This is an equation of a line with a slope of <strong>1</strong>, not <strong>−1</strong>.<br>Choice D is incorrect. This is an equation of a line with a slope of <strong>2</strong>, not <strong>−1</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/c307283c_svg1.svg",
  },
  {
    id: "d62ad380",
    type: "mcq",
    questionHtml:
      "Which of the following could define the relationship between s and P ?",
    choices: [
      { label: "A", text: "<strong>P = 3 s + 10</strong>" },
      { label: "B", text: "<strong>P = ten thirds s + 8</strong>" },
      { label: "C", text: "<strong>P = ten thirds s − 2</strong>" },
      { label: "D", text: "<strong>P = three tenths s − one tenth</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. The relationship between s and P can be modeled by a linear equation of the form P = ks + a, where k and a are constants. The table shows that P increases by 10 when s increases by 3, so k = <strong>the fraction 10 over 3</strong>. To solve for a, substitute one of the given pairs of values for s and P: when s = 3, P = 8, so <strong>8 = the fraction 10 over 3, end fraction · 3 + a</strong>, which yields a = –2. The solution is therefore <strong>P = the fraction 10 over 3, end fraction · s − 2</strong>.Choice A is incorrect. When s = 3, P = 8, but 3(3) + 10 = 19 ≠︀ 8. Choice B is incorrect. This may result from using the first number given for P in the table as the constant term a in the linear equation P = ks + a, which is true only when s = 0. Choice D is incorrect and may result from using the reciprocal of the slope of the line.",
    hasFigure: false,
  },
  {
    id: "f81a0503",
    type: "mcq",
    questionHtml:
      "In the xy-plane, line <strong>k</strong> passes through the points <strong>(0 −5)</strong> and <strong>(1 −1)</strong>. Which equation defines line <strong>k</strong>?",
    choices: [
      { label: "A", text: "<strong>y = −x + one fourth</strong>" },
      { label: "B", text: "<strong>y = one fourth x − 5</strong>" },
      { label: "C", text: "<strong>y = −x + 4</strong>" },
      { label: "D", text: "<strong>y = 4 x − 5</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. An equation defining a line in the xy-plane can be written in the form <strong>y = m x + b</strong>, where <strong>m</strong> represents the slope and <strong>(0, b)</strong> represents the y-intercept of the line. It’s given that line <strong>k</strong> passes through the point <strong>(0 −5)</strong>; therefore, <strong>b = −5</strong>. The slope, <strong>m</strong>, of a line can be found using any two points on the line, <strong>(x 1, y 1)</strong> and <strong>(x 2, y 2)</strong>, and the slope formula <strong>m = (y 2 − y 1) / (x 2 − x 1)</strong>. Substituting the points <strong>(0 −5)</strong> and <strong>(1 −1)</strong> for <strong>(x 1, y 1)</strong> and <strong>(x 2, y 2)</strong>, respectively, in the slope formula yields <strong>m = ((−1 − (−5))) / ((1 − 0))</strong>, or <strong>m = 4</strong>. Substituting <strong>4</strong> for <strong>m</strong> and <strong>−5</strong> for <strong>b</strong> in the equation <strong>y = m x + b</strong> yields <strong>y = 4 x − 5</strong>. <br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "fb43b85f",
    type: "spr",
    questionHtml:
      "A line passes through the points <strong>(4, 6)</strong> and <strong>(15, 24)</strong> in the xy-plane. What is the slope of the line?",
    choices: [],
    correctAnswer: "1.636, 18/11",
    explanation:
      "The correct answer is <strong>(18) / (11)</strong>. For a line that passes through the points <strong>(x 1, y 1)</strong> and <strong>(x 2, y 2)</strong> in the xy-plane, the slope of the line can be calculated using the slope formula, <strong>m = (y 2 − y 1) / (x 2 − x 1)</strong>. It's given that a line passes through the points <strong>(4, 6)</strong> and <strong>(15, 24)</strong> in the xy-plane. Substituting <strong>(4, 6)</strong> for <strong>(x 1, y 1)</strong> and <strong>(15, 24)</strong> for <strong>(x 2, y 2)</strong> in the slope formula, <strong>m = (y 2 − y 1) / (x 2 − x 1)</strong>, yields <strong>m = (24 − 6) / (15 − 4)</strong>, or <strong>m = (18) / (11)</strong>. Therefore, the slope of the line is <strong>(18) / (11)</strong>. Note that 18/11 and 1.636 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
];

export const LINEAR_EQ_TWO_VAR_HARD: PracticeQuestion[] = [
  {
    id: "0366d965",
    type: "spr",
    questionHtml:
      "The table above shows the coordinates of three points on a line in the xy-plane, where k and n are constants. If the slope of the line is 2, what is the value of <strong>k + n</strong> ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 30. The slope of a line can be found by using the slope formula, <strong>the fraction with numerator y sub 2 − y sub 1, and denominator x sub 2 − x sub 1, end fraction</strong>. It’s given that the slope of the line is 2; therefore, <strong>the fraction with numerator y sub 2 − y sub 1, and denominator x sub 2 − x sub 1, end fraction = 2</strong>. According to the table, the points <strong>with coordinates 3, 7</strong> and <strong>with coordinates k, 11</strong> lie on the line. Substituting the coordinates of these points into the equation gives <strong>the fraction with numerator 11 − 7, and denominator k − 3, end fraction = 2</strong>. Multiplying both sides of this equation by <strong>k − 3</strong> gives <strong>11 − 7 = 2 · (k − 3, )</strong>, or <strong>11 − 7 = 2 k − 6</strong>. Solving for k gives <strong>k = 5</strong>. According to the table, the points <strong>with coordinates 3, 7</strong> and <strong>with coordinates 12, n</strong> also lie on the line. Substituting the coordinates of these points into <strong>the fraction with numerator y sub 2 − y sub 1, and denominator x sub 2 − x sub 1, end fraction = 2</strong> gives <strong>the fraction with numerator n − 7, and denominator 12 − 3, end fraction = 2</strong>. Solving for n gives <strong>n = 25</strong>. Therefore, <strong>k + n = 5 + 25</strong>, or 30.",
    hasFigure: false,
  },
  {
    id: "05bb1af9",
    type: "mcq",
    questionHtml:
      "The line slants gradually down from left to right.<br>The line passes through the following points:<br><br>(negative 8 comma 4)<br>(0 comma 2)<br>(8 comma 0)<br><br>The graph of <strong>y = f (x) + 14</strong> is shown. Which equation defines function <strong>f</strong>?",
    choices: [
      { label: "A", text: "<strong>f (x) = −one fourth x − 12</strong>" },
      { label: "B", text: "<strong>f (x) = −one fourth x + 16</strong>" },
      { label: "C", text: "<strong>f (x) = −one fourth x + 2</strong>" },
      { label: "D", text: "<strong>f (x) = −one fourth x − 14</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. An equation for the graph shown can be written in slope-intercept form <strong>y = m x + b</strong>, where <strong>m</strong> is the slope of the graph and its y-intercept is <strong>(0, b)</strong>. Since the y-intercept of the graph shown is <strong>(0, 2)</strong>, the value of <strong>b</strong> is <strong>2</strong>. Since the graph also passes through the point <strong>(4, 1)</strong>, the slope can be calculated as <strong>(1 − 2) / (4 − 0)</strong>, or <strong>−one fourth</strong>. Therefore, the value of <strong>m</strong> is <strong>−one fourth</strong>. Substituting <strong>−one fourth</strong> for <strong>m</strong> and <strong>2</strong> for <strong>b</strong> in the equation <strong>y = m x + b</strong> yields <strong>y = − one fourth x + 2</strong>. It’s given that an equation for the graph shown is <strong>y = f (x) + 14</strong>. Substituting <strong>f (x) + 14</strong> for <strong>y</strong> in the equation <strong>y = − one fourth x + 2</strong>  yields <strong>f (x) + 14 = − one fourth x + 2</strong>. Subtracting <strong>14</strong> from both sides of this equation yields <strong>f (x) = − one fourth x − 12</strong>.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: true,
    figureUrl: "/practice-images/05bb1af9_svg1.svg",
  },
  {
    id: "0b46bad5",
    type: "mcq",
    questionHtml:
      "In the equation above, a and b are constants and <strong>0 < a, which < b</strong>. Which of the following could represent the graph of the equation in the xy-plane?",
    choices: [
      {
        label: "A",
        text: "<strong>The answer choice presents the graph of a line in the xy-plane. The number 1 is indicated on both axes. The line slants downward and to the right, crossing the y axis at 1, and crossing the x axis at 1.</strong>",
      },
      {
        label: "B",
        text: "<strong>The answer choice presents the graph of a line in the xy-plane. The number 1 is indicated on both axes. The line slants downward and to the right, crossing the x axis at − 1, and crossing the y axis at − 1.</strong>",
      },
      {
        label: "C",
        text: "<strong>The answer choice presents the graph of a line in the xy-plane. The number 1 is indicated on both axes. The line slants downward and to the right, crossing the y axis at 1, and crossing the x axis at 2.</strong>",
      },
      {
        label: "D",
        text: "<strong>The answer choice presents the graph of a line in the xy-plane. The number 1 is indicated on both axes. The line slants downward and to the right, crossing the y axis at 1, and crossing the x axis at 0 . 5.</strong>",
      },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. The given equation <strong>a, x + b y = b</strong> can be rewritten in slope-intercept form, <strong>y = m x + k</strong>, where m represents the slope of the line represented by the equation, and k represents the y-coordinate of the y-intercept of the line. Subtracting ax from both sides of the equation yields <strong>b y = −a, x + b</strong>, and dividing both sides of this equation by b yields <strong>y = the − of the fraction a over b, end fraction · x + the fraction b over b, end fraction</strong>, or <strong>y = the − of the fraction a over b, end fraction · x + 1</strong>. With the equation now in slope-intercept form, it shows that <strong>k = 1</strong>, which means the y-coordinate of the y-intercept is 1. It’s given that a and b are both greater than 0 (positive) and that <strong>a < b</strong>. Since <strong>m = the − of the fraction a over b</strong>, the slope of the line must be a value between <strong>−1</strong> and 0. Choice C is the only graph of a line that has a y-value of the y-intercept that is 1 and a slope that is between <strong>−1</strong> and 0.Choices A, B, and D are incorrect because the slopes of the lines in these graphs aren’t between <strong>−1</strong> and 0.",
    hasFigure: true,
    figureUrl: "/practice-images/0b46bad5_img3.png",
  },
  {
    id: "2d54c272",
    type: "spr",
    questionHtml:
      "<strong>5 G + 45 R = 380</strong><br>At a school fair, students can win colored tokens that are worth a different number of points depending on the color. One student won <strong>G</strong> green tokens and <strong>R</strong> red tokens worth a total of <strong>380</strong> points. The given equation represents this situation. How many more points is a red token worth than a green token?",
    choices: [],
    correctAnswer: "40",
    explanation:
      "The correct answer is <strong>40</strong>. It's given that <strong>5 G + 45 R = 380</strong>, where <strong>G</strong> is the number of green tokens and <strong>R</strong> is the number of red tokens won by one student and these tokens are worth a total of <strong>380</strong> points. Since the equation represents the situation where the student won points with green tokens and red tokens for a total of <strong>380</strong> points, each term on the left-hand side of the equation represents the number of points won for one of the colors. Since the coefficient of <strong>G</strong> in the given equation is <strong>5</strong>, a green token must be worth <strong>5</strong> points. Similarly, since the coefficient of <strong>R</strong> in the given equation is <strong>45</strong>, a red token must be worth <strong>45</strong> points. Therefore, a red token is worth <strong>45 − 5</strong> points, or <strong>40</strong> points, more than a green token.",
    hasFigure: false,
  },
  {
    id: "3008cfc3",
    type: "spr",
    questionHtml:
      "<strong>x</strong><br><strong>y</strong><br><br><strong>k</strong><br><strong>13</strong><br><br><strong>k + 7</strong><br><strong>−15</strong><br><br>The table gives the coordinates of two points on a line in the xy-plane. The y-intercept of the line is <strong>(k − 5, b)</strong>, where <strong>k</strong> and <strong>b</strong> are constants. What is the value of <strong>b</strong>?",
    choices: [],
    correctAnswer: "33",
    explanation:
      "The correct answer is <strong>33</strong>. It’s given in the table that the coordinates of two points on a line in the xy-plane are <strong>(k, 13)</strong> and <strong>(k + 7 −15)</strong>. The y-intercept is another point on the line. The slope computed using any pair of points from the line will be the same. The slope of a line, <strong>m</strong>, between any two points, <strong>(x 1, y 1)</strong> and <strong>(x 2, y 2)</strong>, on the line can be calculated using the slope formula, <strong>m = ((y 2 − y 1)) / ((x 2 − x 1))</strong>. It follows that the slope of the line with the given points from the table, <strong>(k, 13)</strong> and <strong>(k + 7 −15)</strong>, is <strong>m = (−15 − 13) / (k + 7 − k)</strong>, which is equivalent to <strong>m = (−28) / (7)</strong>, or <strong>m = −4</strong>. It's given that the y-intercept of the line is <strong>(k − 5, b)</strong>. Substituting <strong>−4</strong> for <strong>m</strong> and the coordinates of the points <strong>(k − 5, b)</strong> and <strong>(k, 13)</strong> into the slope formula yields <strong>−4 = (13 − b) / (k − (k − 5))</strong>, which is equivalent to <strong>−4 = (13 − b) / (k − k + 5)</strong>, or <strong>−4 = (13 − b) / (5)</strong>. Multiplying both sides of this equation by <strong>5</strong> yields <strong>−20 = 13 − b</strong>. Subtracting <strong>13</strong> from both sides of this equation yields <strong>−33 = −b</strong>. Dividing both sides of this equation by <strong>−1</strong> yields <strong>b = 33</strong>. Therefore, the value of <strong>b</strong> is <strong>33</strong>.",
    hasFigure: false,
  },
  {
    id: "3cdbf026",
    type: "mcq",
    questionHtml:
      "The graph of the equation <strong>a, x + k y = 6</strong> is a line in the xy-plane, where a and k are constants. If the line contains the points <strong>with coordinates − 2 −6</strong>and <strong>0 −3</strong>, what is the value of k ?",
    choices: [
      { label: "A", text: "<strong>−2</strong>" },
      { label: "B", text: "<strong>−1</strong>" },
      { label: "C", text: "<strong>2</strong>" },
      { label: "D", text: "<strong>3</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The value of k can be found using the slope-intercept form of a linear equation, <strong>y = m x + b</strong>, where m is the slope and b is the y-coordinate of the y-intercept. The equation <strong>a, x + k y = 6</strong> can be rewritten in the form <strong>y = the − of the fraction a, x, over k, end fraction + the fraction 6 over k</strong>. One of the given points, <strong>with coordinates 0 −3</strong>, is the y-intercept. Thus, the y-coordinate of the y-intercept <strong>−3</strong> must be equal to <strong>the fraction 6 over k</strong>. Multiplying both sides by k gives <strong>−3 k = 6</strong>. Dividing both sides by <strong>−3</strong> gives <strong>k = −2</strong>.Choices B, C, and D are incorrect and may result from errors made rewriting the given equation.",
    hasFigure: false,
  },
  {
    id: "686b7244",
    type: "spr",
    questionHtml:
      "A certain apprentice has enrolled in <strong>85</strong> hours of training courses. The equation <strong>10 x + 15 y = 85</strong> represents this situation, where <strong>x</strong> is the number of on-site training courses and <strong>y</strong> is the number of online training courses this apprentice has enrolled in. How many more hours does each online training course take than each on-site training course?",
    choices: [],
    correctAnswer: "5",
    explanation:
      "The correct answer is <strong>5</strong>. It's given that the equation <strong>10 x + 15 y = 85</strong> represents the situation, where <strong>x</strong> is the number of on-site training courses, <strong>y</strong> is the number of online training courses, and <strong>85</strong> is the total number of hours of training courses the apprentice has enrolled in. Therefore, <strong>10 x</strong> represents the number of hours the apprentice has enrolled in on-site training courses, and <strong>15 y</strong> represents the number of hours the apprentice has enrolled in online training courses. Since <strong>x</strong> is the number of on-site training courses and <strong>y</strong> is the number of online training courses the apprentice has enrolled in, <strong>10</strong> is the number of hours each on-site course takes and <strong>15</strong> is the number of hours each online course takes. Subtracting these numbers gives <strong>15 − 10</strong>, or <strong>5</strong> more hours each online training course takes than each on-site training course.",
    hasFigure: false,
  },
  {
    id: "94b48cbf",
    type: "mcq",
    questionHtml:
      "The graph of <strong>7 x + 2 y = −31</strong> in the <strong>xy</strong>-plane has an <strong>x</strong>-intercept at <strong>(a, 0)</strong> and a <strong>y</strong>-intercept at <strong>(0, b)</strong>, where <strong>a</strong> and <strong>b</strong> are constants. What is the value of <strong>(b) / (a)</strong>?",
    choices: [
      { label: "A", text: "<strong>−seven halves</strong>" },
      { label: "B", text: "<strong>−two sevenths</strong>" },
      { label: "C", text: "<strong>two sevenths</strong>" },
      { label: "D", text: "<strong>seven halves</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The x-coordinate <strong>a</strong> of the x-intercept <strong>(a, 0)</strong> can be found by substituting <strong>0</strong> for <strong>y</strong> in the given equation, which gives <strong>7 x + 2 (0) = −31</strong>, or <strong>7 x = −31</strong>. Dividing both sides of this equation by <strong>7</strong> yields <strong>x = −(31) / (7)</strong>. Therefore, the value of <strong>a</strong> is <strong>−(31) / (7)</strong>. The y-coordinate <strong>b</strong> of the y-intercept <strong>(0, b)</strong> can be found by substituting <strong>0</strong> for <strong>x</strong> in the given equation, which gives <strong>7 (0) + 2 y = −31</strong>, or <strong>2 y = −31</strong>. Dividing both sides of this equation by <strong>2</strong> yields <strong>y = −(31) / (2)</strong>. Therefore, the value of <strong>b</strong> is <strong>−(31) / (2)</strong>. It follows that the value of <strong>(b) / (a)</strong> is <strong>Start(−StartFraction 31) / (2) OverOver − (31) / (7) EndEndFraction</strong>, which is equivalent to <strong>((31) / (2)) (seven thirty firsts)</strong>, or <strong>seven halves</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "98d3393a",
    type: "mcq",
    questionHtml:
      "Line <strong>l</strong> in the xy-plane is perpendicular to the line with equation <strong>x = 2</strong>. What is the slope of line <strong>l</strong>?",
    choices: [
      { label: "A", text: "<strong>0</strong>" },
      { label: "B", text: "<strong>−one half</strong>" },
      { label: "C", text: "<strong>−2</strong>" },
      {
        label: "D",
        text: "The slope of line <strong>l</strong> is undefined.",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It is given that line <strong>l</strong> is perpendicular to a line whose equation is x = 2. A line whose equation is a constant value of x is vertical, so <strong>l</strong> must therefore be horizontal. Horizontal lines have a slope of 0, so <strong>l</strong> has a slope of 0.Choice B is incorrect. A line with slope <strong>−one half</strong> is perpendicular to a line with slope 2. However, the line with equation x = 2 is vertical and has undefined slope (not slope of 2). Choice C is incorrect. A line with slope –2 is perpendicular to a line with slope <strong>one half</strong>. However, the line with equation x = 2 has undefined slope (not slope of <strong>one half</strong>). Choice D is incorrect; this is the slope of the line x = 2 itself, not the slope of a line perpendicular to it.",
    hasFigure: false,
  },
  {
    id: "9bbce683",
    type: "mcq",
    questionHtml:
      "<strong>x</strong><br><strong>y</strong><br><br><strong>18</strong><br><strong>130</strong><br><br><strong>23</strong><br><strong>160</strong><br><br><strong>26</strong><br><strong>178</strong><br><br>For line <strong>h</strong>, the table shows three values of <strong>x</strong> and their corresponding values of <strong>y</strong>. Line <strong>k</strong> is the result of translating line <strong>h</strong> down <strong>5</strong> units in the xy-plane. What is the x-intercept of line <strong>k</strong>?",
    choices: [
      { label: "A", text: "<strong>(−(26) / (3), 0)</strong>" },
      { label: "B", text: "<strong>(−nine halves, 0)</strong>" },
      { label: "C", text: "<strong>(−(11) / (3), 0)</strong>" },
      { label: "D", text: "<strong>(−(17) / (6), 0)</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The equation of line <strong>h</strong> can be written in slope-intercept form <strong>y = m x + b</strong>, where <strong>m</strong> is the slope of the line and <strong>(0, b)</strong> is the y-intercept of the line. It’s given that line <strong>h</strong> contains the points <strong>(18, 130)</strong>, <strong>(23, 160)</strong>, and <strong>(26, 178)</strong>. Therefore, its slope <strong>m</strong> can be found as <strong>(160 − 130) / (23 − 18)</strong>, or <strong>6</strong>. Substituting <strong>6</strong> for <strong>m</strong> in the equation <strong>y = m x + b</strong> yields <strong>y = 6 x + b</strong>. Substituting <strong>130</strong> for <strong>y</strong> and <strong>18</strong> for <strong>x</strong> in this equation yields <strong>130 = 6 (18) + b</strong>, or <strong>130 = 108 + b</strong>. Subtracting <strong>108</strong> from both sides of this equation yields <strong>22 = b</strong>. Substituting <strong>22</strong> for <strong>b</strong> in <strong>y = 6 x + b</strong> yields <strong>y = 6 x + 22</strong>. Since line <strong>k</strong> is the result of translating line <strong>h</strong> down <strong>5</strong> units, an equation of line <strong>k</strong> is <strong>y = 6 x + 22 − 5</strong>, or <strong>y = 6 x + 17</strong>. Substituting <strong>0</strong> for <strong>y</strong> in this equation yields <strong>0 = 6 x + 17</strong>. Solving this equation for <strong>x</strong> yields <strong>x = −(17) / (6)</strong>. Therefore, the x-intercept of line <strong>k</strong> is <strong>(−(17) / (6), 0)</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "9d0396d4",
    type: "mcq",
    questionHtml:
      "The line slants gradually down from left to right.<br>The line passes through the following points:<br><br>(0 comma 7)<br>(8 comma 0)<br><br>The point with coordinates <strong>(d, 4)</strong> lies on the line shown. What is the value of <strong>d</strong>?",
    choices: [
      { label: "A", text: "<strong>seven halves</strong>" },
      { label: "B", text: "<strong>(26) / (7)</strong>" },
      { label: "C", text: "<strong>(24) / (7)</strong>" },
      { label: "D", text: "<strong>(27) / (8)</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It's given from the graph that the points <strong>(0, 7)</strong> and <strong>(8, 0)</strong> lie on the line. For two points on a line, <strong>(x 1, y 1)</strong> and <strong>(x 2, y 2)</strong>, the slope of the line can be calculated using the slope formula <strong>m = (y 2 − y 1) / (x 2 − x 1)</strong>. Substituting <strong>(0, 7)</strong> for <strong>(x 1, y 1)</strong> and <strong>(8, 0)</strong> for <strong>(x 2, y 2)</strong> in this formula, the slope of the line can be calculated as <strong>m = (0 − 7) / (8 − 0)</strong>, or <strong>m = −seven eighths</strong>. It's also given that the point <strong>(d, 4)</strong> lies on the line. Substituting <strong>(d, 4)</strong> for <strong>(x 1, y 1)</strong>, <strong>(8, 0)</strong> for <strong>(x 2, y 2)</strong>, and <strong>−seven eighths</strong> for <strong>m</strong> in the slope formula yields <strong>−seven eighths = (0 − 4) / (8 − d)</strong>, or <strong>−seven eighths = (−4) / (8 − d)</strong>. Multiplying both sides of this equation by <strong>8 − d</strong> yields <strong>− seven eighths (8 − d) = −4</strong>. Expanding the left-hand side of this equation yields <strong>−7 + seven eighths d = −4</strong>. Adding <strong>7</strong> to both sides of this equation yields <strong>seven eighths d = 3</strong>. Multiplying both sides of this equation by <strong>eight sevenths</strong> yields <strong>d = (24) / (7)</strong>. Thus, the value of <strong>d</strong> is <strong>(24) / (7)</strong>.<br>Choice A is incorrect. This is the value of <strong>y</strong> when <strong>x = 4</strong>.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: true,
    figureUrl: "/practice-images/9d0396d4_svg1.svg",
  },
  {
    id: "a1fd2304",
    type: "spr",
    questionHtml:
      "How many liters of a 25% saline solution must be added to 3 liters of a 10% saline solution to obtain a 15% saline solution?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 1.5. The total amount, in liters, of a saline solution can be expressed as the liters of each type of saline solution multiplied by the percent concentration of the saline solution. This gives <strong>3 · 0 . 1 0</strong>, <strong>x · 0 . 2 5</strong>, and <strong>(x + 3, ) · (0 . 1 5, )</strong>, where x is the amount, in liters, of 25% saline solution and 10%, 15%, and 25% are represented as 0.10, 0.15, and 0.25, respectively. Thus, the equation <strong>3 · 0 . 1 0 + 0 . 2 5 x = 0 . 1 5 · (x + 3, )</strong> must be true. Multiplying 3 by 0.10 and distributing 0.15 to <strong>(x + 3, )</strong> yields <strong>0 . 3 0 + 0 . 2 5 x = 0 . 1 5 x + 0 . 4 5</strong>. Subtracting 0.15x and 0.30 from each side of the equation gives <strong>0 . 1 0 x = 0 . 1 5</strong>. Dividing each side of the equation by 0.10 yields <strong>x = 1 . 5</strong>. Note that 1.5 and 3/2 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "a35c7164",
    type: "mcq",
    questionHtml:
      "<strong>5 x + 7 y = 1</strong><br><strong>a x + b y = 1</strong><br>In the given pair of equations, <strong>a</strong> and <strong>b</strong> are constants. The graph of this pair of equations in the xy-plane is a pair of perpendicular lines. Which of the following pairs of equations also represents a pair of perpendicular lines?",
    choices: [
      {
        label: "A",
        text: "<strong>10 x + 7 y = 1</strong><br><strong>a x − 2 b y = 1</strong>",
      },
      {
        label: "B",
        text: "<strong>10 x + 7 y = 1</strong><br><strong>a x + 2 b y = 1</strong>",
      },
      {
        label: "C",
        text: "<strong>10 x + 7 y = 1</strong><br><strong>2 a x + b y = 1</strong>",
      },
      {
        label: "D",
        text: "<strong>5 x − 7 y = 1</strong><br><strong>a x + b y = 1</strong>",
      },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Two lines are perpendicular if their slopes are negative reciprocals, meaning that the slope of the first line is equal to <strong>−1</strong> divided by the slope of the second line. Each equation in the given pair of equations can be written in slope-intercept form, <strong>y = m x + b</strong>, where <strong>m</strong> is the slope of the graph of the equation in the xy-plane and <strong>(0, b)</strong> is the y-intercept. For the first equation, <strong>5 x + 7 y = 1</strong>, subtracting <strong>5 x</strong> from both sides gives <strong>7 y = − 5 x + 1</strong>, and dividing both sides of this equation by <strong>7</strong> gives <strong>y = − five sevenths x + one seventh</strong>. Therefore, the slope of the graph of this equation is <strong>−five sevenths</strong>. For the second equation, <strong>a x + b y = 1</strong>, subtracting <strong>a x</strong> from both sides gives <strong>b y = − a x + 1</strong>, and dividing both sides of this equation by <strong>b</strong> gives <strong>y = − (a) / (b) x + (1) / (b)</strong>. Therefore, the slope of the graph of this equation is <strong>− (a) / (b)</strong>. Since the graph of the given pair of equations is a pair of perpendicular lines, the slope of the graph of the second equation, <strong>− (a) / (b)</strong>, must be the negative reciprocal of the slope of the graph of the first equation, <strong>−five sevenths</strong>. The negative reciprocal of <strong>−five sevenths</strong> is  <strong>(−1) / ((−five sevenths))</strong>, or <strong>seven fifths</strong>. Therefore, <strong>− (a) / (b) = seven fifths</strong>, or <strong>(a) / (b) = −seven fifths</strong>. Similarly, rewriting the equations in choice B in slope-intercept form yields <strong>y = − (10) / (7) x + one seventh</strong> and <strong>y = − (a) / (2 b) x + (1) / (2 b)</strong>. It follows that the slope of the graph of the first equation in choice B is <strong>−(10) / (7)</strong> and the slope of the graph of the second equation in choice B is <strong>− (a) / (2 b)</strong>. Since <strong>(a) / (b) = −seven fifths</strong>, <strong>− (a) / (2 b)</strong> is equal to <strong>− (one half) (−seven fifths)</strong>, or <strong>seven tenths</strong>. Since <strong>seven tenths</strong> is the negative reciprocal of <strong>−(10) / (7)</strong>, the pair of equations in choice B represents a pair of perpendicular lines.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "a7a14e87",
    type: "mcq",
    questionHtml:
      "In the xy-plane, line k is defined by <strong>x + y = 0</strong>. Line j is perpendicular to line k, and the y-intercept of line j is <strong>0, 3</strong>. Which of the following is an equation of line j ?",
    choices: [
      { label: "A", text: "<strong>x + y = 3</strong>" },
      { label: "B", text: "<strong>x + y = −3</strong>" },
      { label: "C", text: "<strong>x − y = 3</strong>" },
      { label: "D", text: "<strong>x − y = −3</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It’s given that line j is perpendicular to line k and that line k is defined by the equation <strong>x + y = 0</strong>. This equation can be rewritten in slope-intercept form, <strong>y = m x + b</strong>, where m represents the slope of the line and b represents the y-coordinate of the y-intercept of the line, by subtracting x from both sides of the equation, which yields <strong>y = −x</strong>. Thus, the slope of line k is <strong>−1</strong>. Since line j and line k are perpendicular, their slopes are opposite reciprocals of each other. Thus, the slope of line j is 1. It’s given that the y-intercept of line j is <strong>0, 3</strong>. Therefore, the equation for line j in slope-intercept form is <strong>y = x + 3</strong>, which can be rewritten as <strong>x − y = −3</strong>.Choices A, B, and C are incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "b9835972",
    type: "spr",
    questionHtml:
      "In the xy-plane, line <strong>script l</strong> passes through the point <strong>(0, 0)</strong> and is parallel to the line represented by the equation <strong>y = 8 x + 2</strong>. If line <strong>script l</strong> also passes through the point <strong>(3, d)</strong>, what is the value of <strong>d</strong>?",
    choices: [],
    correctAnswer: "24",
    explanation:
      "The correct answer is <strong>24</strong>. A line in the xy-plane can be defined by the equation <strong>y = m x + b</strong>, where <strong>m</strong> is the slope of the line and <strong>b</strong> is the y-coordinate of the y-intercept of the line. It's given that line <strong>script l</strong> passes through the point <strong>(0, 0)</strong>. Therefore, the y-coordinate of the y-intercept of line <strong>script l</strong> is <strong>0</strong>. It's given that line <strong>script l</strong> is parallel to the line represented by the equation <strong>y = 8 x + 2</strong>. Since parallel lines have the same slope, it follows that the slope of line <strong>script l</strong> is <strong>8</strong>. Therefore, line <strong>script l</strong> can be defined by an equation in the form <strong>y = m x + b</strong>, where <strong>m = 8</strong> and <strong>b = 0</strong>. Substituting <strong>8</strong> for <strong>m</strong> and <strong>0</strong> for <strong>b</strong> in <strong>y = m x + b</strong> yields the equation <strong>y = 8 x + 0</strong>, or <strong>y = 8 x</strong> . If line <strong>script l</strong> passes through the point <strong>(3, d)</strong>, then when <strong>x = 3</strong>, <strong>y = d</strong> for the equation <strong>y = 8 x</strong>. Substituting <strong>3</strong> for <strong>x</strong> and <strong>d</strong> for <strong>y</strong> in the equation <strong>y = 8 x</strong> yields <strong>d = 8 (3)</strong>, or <strong>d = 24</strong>.",
    hasFigure: false,
  },
  {
    id: "c362c210",
    type: "spr",
    questionHtml:
      "<strong>The figure presents a scatterplot titled “Number of Cornflowers and Wallflowers at Garden Store.” The horizontal axis is labeled “Number of cornflowers, ” and the numbers 0 through 20, in increments of 2, are indicated. The vertical axis is labeled “Number of wallflowers, ” and the numbers 0 through 20, in increments of 2, are indicated. There are 5 data points on the graph. The data represented by the points are as follows. Data . 1. Number of cornflowers, 0. Number of wallflowers, 12.<br>Data . 2. Number of cornflowers, 4. Number of wallflowers, 9.<br>Data . 3. Number of cornflowers, 8. Number of wallflowers, 6.<br>Data . 4. Number of cornflowers, 12. Number of wallflowers, 3.<br>Data . 5. Number of cornflowers, 16. Number of wallflowers, 0.</strong>The points plotted in the coordinate plane above represent the possible numbers of wallflowers and cornflowers that someone can buy at the Garden Store in order to spend exactly $24.00 total on the two types of flowers. The price of each wallflower is the same and the price of each cornflower is the same. What is the price, in dollars, of 1 cornflower?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 1.5. The point <strong>with coordinates 16, 0</strong> corresponds to the situation where 16 cornflowers and 0 wallflowers are purchased. Since the total spent on the two types of flowers is $24.00, it follows that the price of 16 cornflowers is $24.00, and the price of one cornflower is $1.50. Note that 1.5 and 3/2 are examples of ways to enter a correct answer.",
    hasFigure: true,
    figureUrl: "/practice-images/c362c210_img1.png",
  },
  {
    id: "c4ea43ef",
    type: "mcq",
    questionHtml:
      "To earn money for college, Avery works two part-time jobs: A and B. She earns $10 per hour working at job A and $20 per hour working at job B. In one week, Avery earned a total of s dollars for working at the two part-time jobs. The graph above represents all possible combinations of numbers of hours Avery could have worked at the two jobs to earn s dollars. What is the value of s ?",
    choices: [
      { label: "A", text: "128" },
      { label: "B", text: "160" },
      { label: "C", text: "200" },
      { label: "D", text: "320" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Avery earns $10 per hour working at job A. Therefore, if she works a hours at job A, she will earn <strong>10 a</strong> dollars. Avery earns $20 per hour working at job B. Therefore, if she works b hours at job B, she will earn <strong>20 b</strong> dollars. The graph shown represents all possible combinations of the number of hours Avery could have worked at the two jobs to earn s dollars. Therefore, if she worked a hours at job A, worked b hours at job B, and earned s dollars from both jobs, the following equation represents the graph: <strong>10 a + 20 b = s</strong>, where s is a constant. Identifying any point <strong>with coordinates a, , b</strong> from the graph and substituting the values of the coordinates for a and b, respectively, in this equation yield the value of s. For example, the point <strong>with coordinates 16, 0</strong>, where <strong>a = 16</strong> and <strong>b = 0</strong>, lies on the graph. Substituting 16 for a and 0 for b in the equation <strong>10 a + 20 b = s</strong> yields <strong>10 · 16 + 20 · 0 = s</strong>, or <strong>160 = s</strong>. Similarly, the point <strong>with coordinates 0, 8</strong>, where <strong>a = 0</strong> and <strong>b = 8</strong>, lies on the graph. Substituting 0 for a and 8 for b in the equation <strong>10 a + 20 b = s</strong> yields <strong>10 · 0 + 20 · 8 = s</strong>, or <strong>160 = s</strong>.Choices A, C, and D are incorrect. If the value of s is 128, 200, or 320, then no points <strong>with coordinates a, , b</strong> on the graph will satisfy this equation. For example, if the value of s is 128 (choice A), then the equation <strong>10 a + 20 b = s</strong> becomes <strong>10 a + 20 b = 128</strong>. The point <strong>16, 0</strong>, where <strong>a = 16</strong> and <strong>b = 0</strong>, lies on the graph. However, substituting 16 for a and 0 for b in <strong>10 a + 20 b = s</strong> yields <strong>10 · 16 + 20 · 0 = 128</strong>, or <strong>160 = 128</strong>, which is false. Therefore, <strong>16, 0</strong> doesn’t satisfy the equation, and so the value of s can’t be 128. Similarly, if <strong>s = 200</strong> (choice C) or <strong>s = 320</strong> (choice D), then substituting 16 for a and 0 for b yields <strong>160 = 200</strong> and <strong>160 = 320</strong>, respectively, which are both false.",
    hasFigure: true,
    figureUrl: "/practice-images/c4ea43ef_img1.png",
  },
  {
    id: "cb58833c",
    type: "spr",
    questionHtml:
      "The line with the equation <strong>four fifths x + one third y = 1</strong> is graphed in the xy‑plane. What is the x-coordinate of the x‑intercept of the line?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 1.25. The y-coordinate of the x-intercept is 0, so 0 can be substituted for y, giving <strong>four fifths x + one third · 0 = 1</strong>. This simplifies to <strong>four fifths x = 1</strong>. Multiplying both sides of <strong>four fifths x = 1</strong> by 5 gives <strong>4 x = 5</strong>. Dividing both sides of <strong>4 x = 5</strong> by 4 gives <strong>x = five fourths</strong>, which is equivalent to 1.25. Note that 1.25 and 5/4 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "cc3e9528",
    type: "spr",
    questionHtml:
      "The graph of <strong>9 x − 10 y = 19</strong> is translated down <strong>4</strong> units in the xy-plane. What is the x-coordinate of the x-intercept of the resulting graph?",
    choices: [],
    correctAnswer: "59/9, 6.555, 6.556",
    explanation:
      "The correct answer is <strong>(59) / (9)</strong>. When the graph of an equation in the form <strong>A x + B y = C</strong>, where <strong>A</strong>, <strong>B</strong>, and <strong>C</strong> are constants, is translated down <strong>k</strong> units in the xy-plane, the resulting graph can be represented by the equation <strong>A x + B (y + k) = C</strong>. It’s given that the graph of <strong>9 x − 10 y = 19</strong> is translated down <strong>4</strong> units in the xy-plane. Therefore, the resulting graph can be represented by the equation <strong>9 x − 10 (y + 4) = 19</strong>, or <strong>9 x − 10 y − 40 = 19</strong>. Adding <strong>40</strong> to both sides of this equation yields <strong>9 x − 10 y = 59</strong>. The x-coordinate of the x-intercept of the graph of an equation in the xy-plane is the value of <strong>x</strong> in the equation when <strong>y = 0</strong>. Substituting <strong>0</strong> for <strong>y</strong> in the equation <strong>9 x − 10 y = 59</strong> yields <strong>9 x − 10 (0) = 59</strong>, or <strong>9 x = 59</strong>. Dividing both sides of this equation by <strong>9</strong> yields <strong>x = (59) / (9)</strong>. Therefore, the x-coordinate of the x-intercept of the resulting graph is <strong>(59) / (9)</strong>. Note that 59/9, 6.555, and 6.556 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "cc7ffe02",
    type: "mcq",
    questionHtml:
      "Keenan made <strong>32</strong> cups of vegetable broth. Keenan then filled <strong>x</strong> small jars and <strong>y</strong> large jars with all the vegetable broth he made. The equation <strong>3 x + 5 y = 32</strong> represents this situation. Which is the best interpretation of <strong>5 y</strong> in this context?",
    choices: [
      { label: "A", text: "The number of large jars Keenan filled" },
      { label: "B", text: "The number of small jars Keenan filled" },
      {
        label: "C",
        text: "The total number of cups of vegetable broth in the large jars",
      },
      {
        label: "D",
        text: "The total number of cups of vegetable broth in the small jars",
      },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It’s given that the equation <strong>3 x + 5 y = 32</strong> represents the situation where Keenan filled <strong>x</strong> small jars and <strong>y</strong> large jars with all the vegetable broth he made, which was <strong>32</strong> cups. Therefore, <strong>3 x</strong> represents the total number of cups of vegetable broth in the small jars and <strong>5 y</strong> represents the total number of cups of vegetable broth in the large jars. <br>Choice A is incorrect. The number of large jars Keenan filled is represented by <strong>y</strong>, not <strong>5 y</strong>.<br>Choice B is incorrect. The number of small jars Keenan filled is represented by <strong>x</strong>, not <strong>5 y</strong>.<br>Choice D is incorrect. The total number of cups of vegetable broth in the small jars is represented by <strong>3 x</strong>, not <strong>5 y</strong>.",
    hasFigure: false,
  },
  {
    id: "db422e7f",
    type: "spr",
    questionHtml:
      "Line <strong>p</strong> is defined by <strong>4 y + 8 x = 6</strong>. Line <strong>r</strong> is perpendicular to line <strong>p</strong> in the xy-plane. What is the slope of line <strong>r</strong>?",
    choices: [],
    correctAnswer: ".5, 1/2",
    explanation:
      "The correct answer is <strong>one half</strong>. For an equation in slope-intercept form <strong>y = m x + b</strong>, <strong>m</strong> represents the slope of the line in the xy-plane defined by this equation. It's given that line <strong>p</strong> is defined by <strong>4 y + 8 x = 6</strong>. Subtracting <strong>8 x</strong> from both sides of this equation yields <strong>4 y = − 8 x + 6</strong>. Dividing both sides of this equation by <strong>4</strong> yields <strong>y = − eight fourths x + six fourths</strong>, or <strong>y = − 2 x + three halves</strong>. Thus, the slope of line <strong>p</strong> is <strong>−2</strong>. If line <strong>r</strong> is perpendicular to line <strong>p</strong>, then the slope of line <strong>r</strong> is the negative reciprocal of the slope of line <strong>p</strong>. The negative reciprocal of <strong>−2</strong> is <strong>− (1) / ((−2)) = one half</strong>. Note that 1/2 and .5 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "fdee0fbf",
    type: "spr",
    questionHtml:
      "In the xy-plane, line k intersects the y-axis at the point <strong>with coordinates 0 −6</strong> and passes through the point <strong>with coordinates 2, 2</strong>. If the point <strong>with coordinates 20, w</strong> lies on line k, what is the value of w ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 74. The y-intercept of a line in the xy-plane is the ordered pair <strong>x, y</strong> of the point of intersection of the line with the y-axis. Since line k intersects the y-axis at the point <strong>with coordinates 0 −6</strong>, it follows that <strong>0 −6</strong> is the y-intercept of this line. An equation of any line in the xy-plane can be written in the form <strong>y = m x + b</strong>, where m is the slope of the line and b is the y-coordinate of the y-intercept. Therefore, the equation of line k can be written as <strong>y = m x + −6</strong>, or <strong>y = m x − 6</strong>. The value of m can be found by substituting the x- and y-coordinates from a point on the line, such as <strong>2, 2</strong>, for x and y, respectively. This results in <strong>2 = 2 m − 6</strong>. Solving this equation for m gives <strong>m = 4</strong>. Therefore, an equation of line k is <strong>y = 4 x − 6</strong>. The value of w can be found by substituting the x-coordinate, 20, for x in the equation of line k and solving this equation for y. This gives <strong>y = 4 · 20 − 6</strong>, or <strong>y = 74</strong>. Since w is the y-coordinate of this point, <strong>w = 74</strong>.",
    hasFigure: false,
  },
];
