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

export const SYSTEMS_EASY: PracticeQuestion[] = [
  {
    id: "0d1dca87",
    type: "spr",
    questionHtml:
      "If <strong>the ordered pair x, y</strong> is the solution to the given system of equations, what is the value of y ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 23. Since it’s given that <strong>x = 2</strong>, the value of y can be found by substituting 2 for x in the first equation and solving for y. Substituting 2 for x yields <strong>3 · 2 + y = 29</strong>, or <strong>6 + y = 29</strong>. Subtracting 6 from both sides of this equation yields <strong>y = 23</strong>.",
    hasFigure: false,
  },
  {
    id: "0df106df",
    type: "mcq",
    questionHtml:
      "An online bookstore sells novels and magazines. Each novel sells for $4, and each magazine sells for $1. If Sadie purchased a total of 11 novels and magazines that have a combined selling price of $20, how many novels did she purchase?",
    choices: [
      { label: "A", text: "2" },
      { label: "B", text: "3" },
      { label: "C", text: "4" },
      { label: "D", text: "5" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Let n be the number of novels and m be the number of magazines that Sadie purchased. If Sadie purchased a total of 11 novels and magazines, then <strong>n + m = 11</strong>. It is given that the combined price of 11 novels and magazines is $20. Since each novel sells for $4 and each magazine sells for $1, it follows that <strong>4 n + m = 20</strong>. So the system of equations below must hold.<strong>4 n + m = 20; n + m = 11</strong><br>Subtracting corresponding sides of the second equation from the first equation yields <strong>3 n = 9</strong>, so <strong>n = 3</strong>. Therefore, Sadie purchased 3 novels.<br>Choice A is incorrect. If 2 novels were purchased, then a total of $8 was spent on novels. That leaves $12 to be spent on magazines, which means that 12 magazines would have been purchased. However, Sadie purchased a total of 11 novels and magazines. Choices C and D are incorrect. If 4 novels were purchased, then a total of $16 was spent on novels. That leaves $4 to be spent on magazines, which means that 4 magazines would have been purchased. By the same logic, if Sadie purchased 5 novels, she would have no money at all ($0) to buy magazines. However, Sadie purchased a total of 11 novels and magazines.",
    hasFigure: false,
  },
  {
    id: "17f176ec",
    type: "mcq",
    questionHtml:
      "A movie theater charges $11 for each full-price ticket and $8.25 for each reduced-price ticket. For one movie showing, the theater sold a total of 214 full-price and reduced-price tickets for $2,145. Which of the following systems of equations could be used to determine the number of full-price tickets, f, and the number of reduced-price tickets, r, sold?",
    choices: [
      {
        label: "A",
        text: "<strong>f + r = 2, 145, and, 11 f + 8 . 2 5 r = 214</strong>",
      },
      {
        label: "B",
        text: "<strong>f + r = 214, and, 11 f + 8 . 2 5 r = 2, 145</strong>",
      },
      {
        label: "C",
        text: "<strong>f + r = 214, and, 8 . 2 5 f + 11 r = 2, 145</strong>",
      },
      {
        label: "D",
        text: "<strong>f + r = 2, 145, and, 8 . 2 5 f + 11 r = 214</strong>",
      },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The movie theater sells f full-price tickets and r reduced-price tickets, so the total number of tickets sold is f + r. Since the movie theater sold a total of 214 full-price and reduced-price tickets for one movie showing, it follows that f + r = 214. The movie theater charges $11 for each full-price ticket; thus, the sales for full-price tickets, in dollars, is given by 11f. The movie theater charges $8.25 for each reduced-price ticket; thus, the sales for reduced-price tickets, in dollars, is given by 8.25r. Therefore, the total sales, in dollars, for the movie showing is given by 11f + 8.25r. Since the total sales for all full-price and reduced-price tickets is $2,145, it follows that 11f + 8.25r = 2,145.Choice A is incorrect. This system of equations suggests that the movie theater sold a total of 2,145 full-price and reduced-price tickets for a total of $214. Choice C is incorrect. This system suggests that the movie theater charges $8.25 for each full-price ticket and $11 for each reduced-price ticket. Choice D is incorrect. This system suggests that the movie theater charges $8.25 for each full-price ticket and $11 for each reduced-price ticket and sold a total of 2,145 tickets for a total of $214.",
    hasFigure: false,
  },
  {
    id: "317e80f9",
    type: "mcq",
    questionHtml:
      "<strong>x + y = 18</strong><br><strong>5 y = x</strong><br>What is the solution <strong>(x, y)</strong> to the given system of equations?",
    choices: [
      { label: "A", text: "<strong>(15, 3)</strong>" },
      { label: "B", text: "<strong>(16, 2)</strong>" },
      { label: "C", text: "<strong>(17, 1)</strong>" },
      { label: "D", text: "<strong>(18, 0)</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The second equation in the given system defines the value of <strong>x</strong> as <strong>5 y</strong>. Substituting <strong>5 y</strong> for <strong>x</strong> into the first equation yields <strong>5 y + y = 18</strong> or <strong>6 y = 18</strong>. Dividing each side of this equation by <strong>6</strong> yields <strong>y = 3</strong>. Substituting <strong>3</strong> for <strong>y</strong> in the second equation yields <strong>5 (3) = x</strong> or <strong>x = 15</strong>. Therefore, the solution <strong>(x, y)</strong> to the given system of equations is <strong>(15, 3)</strong>. <br>Choice B is incorrect. Substituting <strong>16</strong> for <strong>x</strong> and <strong>2</strong> for <strong>y</strong> in the second equation yields <strong>5 (2) = 16</strong>, which is not true. Therefore, <strong>(16, 2)</strong> is not a solution to the given system of equations.<br>Choice C is incorrect. Substituting <strong>17</strong> for <strong>x</strong> and <strong>1</strong> for <strong>y</strong> in the second equation yields <strong>5 (1) = 17</strong>, which is not true. Therefore, <strong>(17, 1)</strong> is not a solution to the given system of equations.<br>Choice D is incorrect. Substituting <strong>18</strong> for <strong>x</strong> and <strong>0</strong> for <strong>y</strong> in the second equation yields <strong>5 (0) = 18</strong>, which is not true. Therefore, <strong>(18, 0)</strong> is not a solution to the given system of equations.",
    hasFigure: false,
  },
  {
    id: "44d65912",
    type: "mcq",
    questionHtml:
      "Which system of equations can be used to correctly determine the values of c and s ?",
    choices: [
      {
        label: "A",
        text: "<strong>This answer choice consists of two equations. 10 c + 20 s = 700, and, 12 c + 25 s = 850</strong>",
      },
      {
        label: "B",
        text: "<strong>This answer choice consists of two equations. 20 c + 10 s = 700, and, 25 c + 12 s = 850</strong>",
      },
      {
        label: "C",
        text: "<strong>This answer choice consists of two equations. 20 c + 700 s = 10, and, 25 c + 850 s = 12</strong>",
      },
      {
        label: "D",
        text: "<strong>This answer choice consists of two equations. 700 c + 20 s = 10, and, 850 c + 25 s = 12</strong>",
      },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The number of coins collected can be multiplied by c to give the score from the points earned from coins. Similarly, the number of stars collected can be multiplied by s to give the score from the points earned from the stars. Therefore, the total score each time Angela played is <strong>20 c + 10 s = 700</strong>, and the total score the second time she played is <strong>25 c + 12 s = 850</strong>.Choices A, C, and D are incorrect and may result from misidentifying the terms of the equation. Choice A switches coins and stars, choice C switches stars and points, and choice D misidentifies coins, stars, and points.",
    hasFigure: false,
  },
  {
    id: "4b76c7f1",
    type: "mcq",
    questionHtml:
      "In the given system of equations, a is a constant. If the system has infinitely many solutions, what is the value of a ?",
    choices: [
      { label: "A", text: "4" },
      { label: "B", text: "9" },
      { label: "C", text: "36" },
      { label: "D", text: "54" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. A system of two linear equations has infinitely many solutions if one equation is equivalent to the other. This means that when the two equations are written in the same form, each coefficient or constant in one equation is equal to the corresponding coefficient or constant in the other equation multiplied by the same number. The equations in the given system of equations are written in the same form, with x and y on the left-hand side of the equation and a constant on the right-hand side of the equation. The coefficients of x and y in the second equation are equal to the coefficients of x and y, respectively, in the first equation multiplied by 4: <strong>8 = 2 · 4</strong> and <strong>28 = 7 · 4</strong>. Therefore, the constant in the second equation must be equal to 4 times the constant in the first equation: <strong>a = 9 · 4</strong>, or <strong>a = 36</strong>.Choices A, B, and D are incorrect. When <strong>a = 4</strong>, <strong>a = 9</strong>, or <strong>a = 54</strong>, the given system of equations has no solution.",
    hasFigure: false,
  },
  {
    id: "4ec95eab",
    type: "mcq",
    questionHtml:
      "<strong>y = −3 x</strong><br><strong>4 x + y = 15</strong><br>The solution to the given system of equations is <strong>(x, y)</strong>. What is the value of <strong>x</strong>?",
    choices: [
      { label: "A", text: "<strong>1</strong>" },
      { label: "B", text: "<strong>5</strong>" },
      { label: "C", text: "<strong>15</strong>" },
      { label: "D", text: "<strong>45</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. The given system of linear equations can be solved by the substitution method. Substituting <strong>− 3 x</strong> for <strong>y</strong> from the first equation in the given system into the second equation yields <strong>4 x + (−3 x) = 15</strong>, or <strong>x = 15</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 D is incorrect. This is the absolute value of <strong>y</strong>, not the value of <strong>x</strong>.",
    hasFigure: false,
  },
  {
    id: "608eeb6e",
    type: "mcq",
    questionHtml:
      "<strong>5 x = 15</strong><br><strong>− 4 x + y = −2</strong><br>The solution to the given system of equations is <strong>(x, y)</strong>. What is the value of <strong>x + y</strong>?",
    choices: [
      { label: "A", text: "<strong>−17</strong>" },
      { label: "B", text: "<strong>−13</strong>" },
      { label: "C", text: "<strong>13</strong>" },
      { label: "D", text: "<strong>17</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Adding the second equation of the given system to the first equation yields <strong>5 x + (−4 x + y) = 15 + (−2)</strong>, which is equivalent to <strong>x + y = 13</strong>. So the value of <strong>x + y</strong> is <strong>13</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect. This is the value of <strong>− (x + y)</strong>.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "7d89376f",
    type: "mcq",
    questionHtml:
      "A discount airline sells a certain number of tickets, x, for a flight for $90 each. It sells the number of remaining tickets, y, for $250 each. For a particular flight, the airline sold 120 tickets and collected a total of $27,600 from the sale of those tickets. Which system of equations represents this relationship between x and y ?",
    choices: [
      {
        label: "A",
        text: "<strong>open brace, x + y = 120, and, 90 x + 250 y = 27, 600</strong>",
      },
      {
        label: "B",
        text: "<strong>open brace, x + y = 120, and, 90 x + 250 y = 120 · 27, 600</strong>",
      },
      {
        label: "C",
        text: "<strong>open brace, x + y = 27, 600, and, 90 x + 250 y = 120 · 27, 600</strong>",
      },
      {
        label: "D",
        text: "<strong>open brace, 90 x = 250 y, and, 120 x + 120 y = 27, 600</strong>",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The airline sold two types of tickets for this flight: x tickets at $90 each and the remaining  tickets, y, at $250 each. Because the airline sold a total of 120 tickets for this flight, it must be true that x + y = 120. The amount, in dollars, collected from the sale of x tickets at $90 each is represented by 90x. The amount, in dollars, collected from the sale of the remaining y tickets at $250 each is represented by 250y. It is given that a total of $27,600 was collected from the sale of all tickets. Therefore, it must also be true that 90x + 250y = 27,600.Choice B is incorrect. The total number of tickets sold is represented correctly as x + y = 120. The total amount, in dollars, collected from the sale of the x tickets at $90 each and the remaining tickets, y, at $250 has been correctly represented as 90x + 250y. However, according to the information given, this total should be equal to 27,600, not 120(27,600) dollars. Choice C is incorrect. The total number of tickets sold has been correctly represented as x + y. However, according to the information given, this total should be equal to 120, not 27,600, as shown in choice C. The total amount, in dollars, collected from the sale of the x tickets at $90 each and the remaining tickets, y, at $250 has been correctly represented as 90x + 250y. However, according to the information given, this total should be equal to 27,600, not 120(27,600) dollars. Choice D is incorrect. The two equations given in choice D have no meaning in this context.",
    hasFigure: false,
  },
  {
    id: "8abed0fb",
    type: "mcq",
    questionHtml:
      "What is the solution <strong>the ordered pair, x, y, </strong> to the given system of equations?",
    choices: [
      { label: "A", text: "<strong>1, 2</strong>" },
      { label: "B", text: "<strong>1, 5</strong>" },
      { label: "C", text: "<strong>2, 3</strong>" },
      { label: "D", text: "<strong>2, 7</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Since it’s given that <strong>x = 1</strong>, substituting 1 for x in the first equation yields <strong>y = 2 · 1 + 3</strong>. Simplifying the right-hand side of this equation yields <strong>y = 2 + 3</strong>, or <strong>y = 5</strong>. Therefore, the ordered pair <strong>1, 5</strong> is a solution to the given system of equations.Choice A is incorrect and may result from a calculation error when substituting 1 for x in the first equation. Choices C and D are incorrect. Because it’s given that <strong>x = 1</strong>, x cannot equal 2 as stated in these ordered pairs.",
    hasFigure: false,
  },
  {
    id: "9db5b5c1",
    type: "mcq",
    questionHtml:
      "<strong>4 x = 20</strong><br><strong>− 3 x + y = −7</strong><br>The solution to the given system of equations is <strong>(x, y)</strong>. What is the value of <strong>x + y</strong>?",
    choices: [
      { label: "A", text: "<strong>−27</strong>" },
      { label: "B", text: "<strong>−13</strong>" },
      { label: "C", text: "<strong>13</strong>" },
      { label: "D", text: "<strong>27</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It's given that <strong>4 x = 20</strong> and <strong>− 3 x + y = −7</strong> is a system of equations with a solution <strong>(x, y)</strong>. Adding the second equation in the given system to the first equation yields <strong>4 x + (−3 x + y) = 20 + (−7)</strong>, which is equivalent to <strong>x + y = 13</strong>. Thus, the value of <strong>x + y</strong> is <strong>13</strong>. <br>Choice A is incorrect. This represents the value of <strong>− 2 (x + y) − 1</strong>.<br>Choice B is incorrect. This represents the value of <strong>− (x + y)</strong>.<br>Choice D is incorrect. This represents the value of <strong>2 (x + y) + 1</strong>.",
    hasFigure: false,
  },
  {
    id: "aff28230",
    type: "mcq",
    questionHtml:
      "<strong>x = 10</strong><br><strong>y = x + 21</strong><br>The solution to the given system of equations is <strong>(x, y)</strong>. What is the value of <strong>y</strong>?",
    choices: [
      { label: "A", text: "<strong>2.1</strong>" },
      { label: "B", text: "<strong>10</strong>" },
      { label: "C", text: "<strong>21</strong>" },
      { label: "D", text: "<strong>31</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It's given by the first equation in the given system of equations that <strong>x = 10</strong>. Substituting <strong>10</strong> for <strong>x</strong> in the second equation in the given system yields <strong>y = 10 + 21</strong>, or <strong>y = 31</strong>. Therefore, the value of <strong>y</strong> is <strong>31</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect. This is the value of <strong>x</strong>, not the value of <strong>y</strong>.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "b0fc3166",
    type: "mcq",
    questionHtml:
      "For the first line in the system:<br><br>The line is horizontal.<br>The line passes through the following points:<br><br>(negative 4 comma 3)<br>(0 comma 3)<br>(2 comma 3)<br><br>For the second line in the system:<br><br>The line slants sharply up from left to right.<br>The line passes through the following points:<br><br>(negative 1 comma negative 4.5)<br>(0 comma negative 2)<br>(2 comma 3)<br><br>The graph of a system of linear equations is shown. What is the solution <strong>(x, y)</strong> to the system?",
    choices: [
      { label: "A", text: "<strong>(0, 3)</strong>" },
      { label: "B", text: "<strong>(1, 3)</strong>" },
      { label: "C", text: "<strong>(2, 3)</strong>" },
      { label: "D", text: "<strong>(3, 3)</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. The solution to this system of linear equations is represented by the point that lies on both lines shown, or the point of intersection of the two lines. According to the graph, the point of intersection occurs when <strong>x = 2</strong> and <strong>y = 3</strong>, or at the point <strong>(2, 3)</strong>. Therefore, the solution <strong>(x, y)</strong> to the system is <strong>(2, 3)</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 D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: true,
    figureUrl: "/practice-images/b0fc3166_svg1.svg",
  },
  {
    id: "b86123af",
    type: "mcq",
    questionHtml:
      "Hiro and Sofia purchased shirts and pants from a store. The price of each shirt purchased was the same and the price of each pair of pants purchased was the same. Hiro purchased 4 shirts and 2 pairs of pants for $86, and Sofia purchased 3 shirts and 5 pairs of pants for $166. Which of the following systems of linear equations represents the situation, if x represents the price, in dollars, of each shirt and y represents the price, in dollars, of each pair of pants?",
    choices: [
      {
        label: "A",
        text: "<strong>4 x + 2 y = 86, and, 3 x + 5 y = 166</strong>",
      },
      {
        label: "B",
        text: "<strong>4 x + 3 y = 86, and, 2 x + 5 y = 166</strong>",
      },
      {
        label: "C",
        text: "<strong>4 x + 2 y = 166, and, 3 x + 5 y = 86</strong>",
      },
      {
        label: "D",
        text: "<strong>4 x + 3 y = 166, and, 2 x + 5 y = 86</strong>",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Hiro purchased 4 shirts and each shirt cost x dollars, so he spent a total of 4x dollars on shirts. Likewise, Hiro purchased 2 pairs of pants, and each pair of pants cost y dollars, so he spent a total of 2y dollars on pants. Therefore, the total amount that Hiro spent was 4x + 2y. Since Hiro spent $86 in total, this can be modeled by the equation 4x + 2y = 86. Using the same reasoning, Sofia bought 3 shirts at x dollars each and 5 pairs of pants at y dollars each, so she spent a total of 3x + 5y dollars on shirts and pants. Since Sofia spent $166 in total, this can be modeled by the equation 3x + 5y = 166.Choice B is incorrect and may be the result of switching the number of shirts Sofia purchased with the number of pairs of pants Hiro purchased. Choice C is incorrect and may be the result of switching the total price each person paid. Choice D is incorrect and may be the result of switching the total price each person paid as well as switching the number of shirts Sofia purchased with the number of pairs of pants Hiro purchased.",
    hasFigure: false,
  },
  {
    id: "ca9bb527",
    type: "mcq",
    questionHtml:
      "<strong>y = 4 x − 9</strong><br><strong>y = 19</strong><br>What is the solution <strong>(x, y)</strong> to the given system of equations?",
    choices: [
      { label: "A", text: "<strong>(4, 19)</strong>" },
      { label: "B", text: "<strong>(7, 19)</strong>" },
      { label: "C", text: "<strong>(19, 4)</strong>" },
      { label: "D", text: "<strong>(19, 7)</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It's given by the second equation in the system that <strong>y = 19</strong>. Substituting <strong>19</strong> for <strong>y</strong> in the first equation yields <strong>19 = 4 x − 9</strong>. Adding <strong>9</strong> to both sides of this equation yields <strong>28 = 4 x</strong>. Dividing both sides of this equation by <strong>4</strong> yields <strong>7 = x</strong>. Therefore, since <strong>x = 7</strong> and <strong>y = 19</strong>, the solution <strong>(x, y)</strong> to the given system of equations is <strong>(7, 19)</strong>.<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: "cd33b015",
    type: "mcq",
    questionHtml:
      "If <strong>the ordered pair x, y</strong> is the solution to the given system of equations, what is the value of y ?",
    choices: [
      { label: "A", text: "10" },
      { label: "B", text: "15" },
      { label: "C", text: "60" },
      { label: "D", text: "65" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Substituting 20 for <strong>x + y</strong> in the second equation in the system yields <strong>2 · 20 + 3y = 85</strong>, or <strong>40 + 3 y = 85</strong>. Subtracting 40 from both sides of this equation yields <strong>3 y = 45</strong>. Dividing both sides of this equation by 3 yields <strong>y = 15</strong>.Choice A is incorrect. If <strong>y = 10</strong>, then <strong>x = 10</strong> since <strong>x + y = 20</strong>. However, substituting 10 for both x and y in the second equation yields <strong>70 = 85</strong>, which is a false statement. Choice C is incorrect. If <strong>y = 60</strong>, then <strong>x = −40</strong> since <strong>x + y = 20</strong>. However, substituting these values for x and y in the second equation yields <strong>220 = 85</strong>, which is a false statement. Choice D is incorrect. If <strong>y = 65</strong>, then <strong>x = −45</strong> since <strong>x + y = 20</strong>. However, substituting these values for x and y in the second equation yields <strong>235 = 85</strong>, which is a false statement.",
    hasFigure: false,
  },
  {
    id: "dba8d38a",
    type: "mcq",
    questionHtml:
      "A petting zoo sells two types of tickets. The standard ticket, for admission only, costs $5. The premium ticket, which includes admission and food to give to the animals, costs $12. One Saturday, the petting zoo sold a total of 250 tickets and collected a total of $2,300 from ticket sales. Which of the following systems of equations can be used to find the number of standard tickets, s, and premium tickets, p, sold on that Saturday?",
    choices: [
      {
        label: "A",
        text: "<strong>Equation 1: s + p = 250. Equation 2: 5 s + 12 p = 2, 300.</strong>",
      },
      {
        label: "B",
        text: "<strong>Equation 1: s + p = 250. Equation 2: 12 s + 5 p = 2, 300.</strong>",
      },
      {
        label: "C",
        text: "<strong>Equation 1: 5 s + 12 p = 250. Equation 2: s + p = 2, 300.</strong>",
      },
      {
        label: "D",
        text: "<strong>Equation 1: 12 s + 5 p = 250. Equation 2: s + p = 2, 300.</strong>",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It’s given that the petting zoo sells two types of tickets, standard and premium, and that s represents the number of standard tickets sold and p represents the number of premium tickets sold. It’s also given that the petting zoo sold 250 tickets on one Saturday; thus, <strong>s + p = 250</strong>. It’s also given that each standard ticket costs $5 and each premium ticket costs $12. Thus, the amount collected in ticket sales can be represented by <strong>5 s</strong> for standard tickets and <strong>12 p</strong> for premium tickets. On that Saturday the petting zoo collected a total of $2,300 from ticket sales; thus, <strong>5 s + 12 p = 2, 300</strong>. These two equations are correctly represented in choice A.Choice B is incorrect. The second equation in the system represents the cost per standard ticket as $12, not $5, and the cost per premium ticket as $5, not $12. Choices C and D are incorrect. The equations represent the total collected from standard and premium ticket sales as $250, not $2,300, and the total number of standard and premium tickets sold as $2,300, not $250. Additionally, the first equation in choice D represents the cost per standard ticket as $12, not $5, and the cost per premium ticket as $5, not $12.",
    hasFigure: false,
  },
  {
    id: "e1259a5a",
    type: "mcq",
    questionHtml:
      "Which of the following points is the solution to the system of equations?",
    choices: [
      { label: "A", text: "<strong>3, 9</strong>" },
      { label: "B", text: "<strong>6, 15</strong>" },
      { label: "C", text: "<strong>8, 10</strong>" },
      { label: "D", text: "<strong>12, 18</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The solution to this system of linear equations is the point that lies on both lines graphed, or the point of intersection of the two lines. According to the graphs, the point of intersection occurs when <strong>x = 3</strong> and <strong>y = 9</strong>, or at the point <strong>with coordinates 3, 9</strong>.Choices B and D are incorrect. Each of these points lies on one line, but not on both lines in the xy-plane. Choice C is incorrect. This point doesn’t lie on either of the lines graphed in the xy-plane.",
    hasFigure: true,
    figureUrl: "/practice-images/e1259a5a_img1.png",
  },
  {
    id: "ece00725",
    type: "mcq",
    questionHtml:
      "Connor has <strong>c</strong> dollars and Maria has <strong>m</strong> dollars. Connor has <strong>4</strong> times as many dollars as Maria, and together they have a total of <strong>dollar sign 25.00</strong>. Which system of equations represents this situation?",
    choices: [
      {
        label: "A",
        text: "<strong>c = 4 m</strong><br><strong>c + m = 25</strong>",
      },
      {
        label: "B",
        text: "<strong>m = 4 c</strong><br><strong>c + m = 25</strong>",
      },
      {
        label: "C",
        text: "<strong>c = 25 m</strong><br><strong>c + m = 4</strong>",
      },
      {
        label: "D",
        text: "<strong>m = 25 c</strong><br><strong>c + m = 4</strong>",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It’s given that Connor has <strong>c</strong> dollars, Maria has <strong>m</strong> dollars, and Connor has <strong>4</strong> times as many dollars as Maria. This can be represented by the equation <strong>c = 4 m</strong>. It’s also given that together, Connor and Maria have a total of <strong>dollar sign 25.00</strong>, which can be represented by the equation <strong>c + m = 25</strong>. Therefore, the system consisting of the equations <strong>c = 4 m</strong> and <strong>c + m = 25</strong> represents this situation.<br>Choice B is incorrect. The equation <strong>m = 4 c</strong> represents a situation where Maria has <strong>4</strong> times as many dollars as Connor, rather than the situation where Connor has <strong>4</strong> times as many dollars as Maria.<br>Choice C is incorrect. The equation <strong>c = 25 m</strong> represents a situation where Connor has <strong>25</strong> times, rather than <strong>4</strong> times, as many dollars as Maria. The equation <strong>c + m = 4</strong> represents a situation where Connor and Maria together have a total of <strong>dollar sign 4.00</strong>, rather than <strong>dollar sign 25.00</strong>.<br>Choice D is incorrect. The equation <strong>m = 25 c</strong> represents a situation where Maria has <strong>25</strong> times as many dollars as Connor, rather than the situation where Connor has <strong>4</strong> times as many dollars as Maria. The equation <strong>c + m = 4</strong> represents a situation where Connor and Maria together have a total of <strong>dollar sign 4.00</strong>, rather than <strong>dollar sign 25.00</strong>.",
    hasFigure: false,
  },
  {
    id: "ee031767",
    type: "mcq",
    questionHtml:
      "A dance teacher ordered outfits for students for a dance recital. Outfits for boys cost $26, and outfits for girls cost $35. The dance teacher ordered a total of 28 outfits and spent $881. If b represents the number of outfits the dance teacher ordered for boys and g represents the number of outfits the dance teacher ordered for girls, which of the following systems of equations can be solved to find b and g ?",
    choices: [
      {
        label: "A",
        text: "<strong>Each option consists of two equations. 26 b + 35 g = 28, and, b + g = 881</strong>",
      },
      {
        label: "B",
        text: "<strong>26 b + 35 g = 881, and, b + g = 28</strong>",
      },
      {
        label: "C",
        text: "<strong>26 g + 35 b = 28, and, b + g = 881</strong>",
      },
      {
        label: "D",
        text: "<strong>26 g + 35 b = 881, and, b + g = 28</strong>",
      },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Outfits for boys cost $26 each and the teacher ordered b outfits for boys, so the teacher spent 26b dollars on outfits for boys. Similarly, outfits for girls cost $35 each and the teacher ordered g outfits for girls, so the teacher spent 35g dollars on outfits for girls. Since the teacher spent a total of $881 on outfits for boys and girls, the equation 26b + 35g = 881 must be true. And since the teacher ordered a total of 28 outfits, the equation b + g = 28 must also be true.<br> <br><br>Choice A is incorrect and may result from switching the constraint on the total number of outfits with the constraint on the cost of the outfits. Choice C is incorrect and may result from switching the constraint on the total number of outfits with the constraint on the cost of the outfits, as well as switching the cost of the outfits for boys with the cost of the outfits for girls. Choice D is incorrect and may result from switching the cost of the outfits for boys with the cost of the outfits for girls.",
    hasFigure: false,
  },
  {
    id: "f88970cc",
    type: "mcq",
    questionHtml:
      "<strong>x = 5</strong><br><strong>y = x − 8</strong><br>Which of the following points <strong>(x, y)</strong> is the solution to the given system of equations in the xy-plane?",
    choices: [
      { label: "A", text: "<strong>(0, 0)</strong>" },
      { label: "B", text: "<strong>(5 −3)</strong>" },
      { label: "C", text: "<strong>(5 −8)</strong>" },
      { label: "D", text: "<strong>(5, 8)</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. A solution to a system of equations in the xy-plane is a point <strong>(x, y)</strong> that lies on the graph of each equation in the system. The first equation given is <strong>x = 5</strong>. Substituting <strong>5</strong> for <strong>x</strong> in the second given equation yields <strong>y = 5 − 8</strong>, or <strong>y = −3</strong>. It follows that in the xy-plane, the point <strong>(5 −3)</strong> lies on the graph of each equation in the system. Therefore, the solution to the given system of equations in the xy-plane is <strong>(5 −3)</strong>.<br>Choice A is incorrect. The point <strong>(0, 0)</strong> doesn't lie on the graph of either equation in the given system.<br>Choice C is incorrect. The point <strong>(5 −8)</strong> doesn't lie on the graph of the second equation in the given system.<br>Choice D is incorrect. The point <strong>(5, 8)</strong> doesn't lie on the graph of the second equation in the given system.",
    hasFigure: false,
  },
  {
    id: "ffb371f5",
    type: "mcq",
    questionHtml:
      "<strong>3 x = 12</strong><br><strong>−3 x + y = −6</strong><br>The solution to the given system of equations is <strong>(x, y)</strong>. What is the value of <strong>y</strong>?",
    choices: [
      { label: "A", text: "<strong>−3</strong>" },
      { label: "B", text: "<strong>6</strong>" },
      { label: "C", text: "<strong>18</strong>" },
      { label: "D", text: "<strong>30</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Adding the second equation in the given system to the first equation in the given system yields <strong>3 x + (−3 x + y) = 12 + (−6)</strong>, which is equivalent to <strong>y = 6</strong>. <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,
  },
];

export const SYSTEMS_MEDIUM: PracticeQuestion[] = [
  {
    id: "092ad67d",
    type: "mcq",
    questionHtml:
      "<strong>x + 2 y = 6</strong><br><strong>x − 2 y = 4</strong><br>The solution to the given system of equations is <strong>(x, y)</strong>. What is the value of <strong>x</strong>?",
    choices: [
      { label: "A", text: "<strong>2.5</strong>" },
      { label: "B", text: "<strong>5</strong>" },
      { label: "C", text: "<strong>6</strong>" },
      { label: "D", text: "<strong>10</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Adding the first equation to the second equation in the given system yields <strong>(x + 2 y) + (x − 2 y) = 6 + 4</strong>, or <strong>(x + x) + (2 y − 2 y) = 10</strong>. Combining like terms in this equation yields <strong>2 x = 10</strong>. Dividing both sides of this equation by <strong>2</strong> yields <strong>x = 5</strong>. Thus, the value of <strong>x</strong> is <strong>5</strong>.<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. This is the value of <strong>2 x</strong>, not <strong>x</strong>.",
    hasFigure: false,
  },
  {
    id: "0dd6227f",
    type: "mcq",
    questionHtml:
      "At how many points do the graphs of the equations <strong>y = x + 20</strong> and <strong>y = 8 x</strong> intersect in the xy-plane?",
    choices: [
      { label: "A", text: "<strong>0</strong>" },
      { label: "B", text: "<strong>1</strong>" },
      { label: "C", text: "<strong>2</strong>" },
      { label: "D", text: "<strong>8</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Each given equation is written in slope-intercept form, <strong>y = m x + b</strong>, where <strong>m</strong> is the slope and <strong>(0, b)</strong> is the y-intercept of the graph of the equation in the xy-plane. The graphs of two lines that have different slopes will intersect at exactly one point. The graph of the first equation is a line with slope <strong>1</strong>. The graph of the second equation is a line with slope <strong>8</strong>. Since the graphs are lines with different slopes, they will intersect at exactly one point.<br>Choice A is incorrect because two graphs of linear equations have <strong>0</strong> intersection points only if they are parallel and therefore have the same slope.<br>Choice C is incorrect because two graphs of linear equations in the xy-plane can have only <strong>0</strong>, <strong>1</strong>, or infinitely many points of intersection.<br>Choice D is incorrect because two graphs of linear equations in the xy-plane can have only <strong>0</strong>, <strong>1</strong>, or infinitely many points of intersection.",
    hasFigure: false,
  },
  {
    id: "19fdf387",
    type: "mcq",
    questionHtml:
      "In the xy-plane, the graph of <strong>y = x + 3</strong> intersects the graph of <strong>y = 2 x − 6</strong> at the point <strong>with coordinates a, , b</strong>. What is the value of a ?",
    choices: [
      { label: "A", text: "3" },
      { label: "B", text: "6" },
      { label: "C", text: "9" },
      { label: "D", text: "12" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Since the graph of <strong>y = x + 3</strong> intersects the graph of <strong>y = 2 x − 6</strong> at the point <strong>with coordinates a, , b</strong>, the ordered pair <strong>a, , b</strong> is the solution to the system of linear equations consisting of <strong>y = x + 3</strong> and <strong>y = 2 x − 6</strong>, and the value of a is the value of x in the solution of this system. Since both <strong>x + 3</strong>and <strong>2 x − 6</strong> are equal to y, it follows that <strong>x + 3 = 2 x − 6</strong>. Subtracting x from and adding 6 to both sides of the equation yields <strong>9 = x</strong>. Therefore, the value of a is 9.Choices A and B are incorrect and may result from a calculation or conceptual error in solving the system of equations consisting of <strong>y = x + 3</strong> and <strong>y = 2 x − 6</strong>. Choice D is incorrect. This is the value of b, not a.",
    hasFigure: false,
  },
  {
    id: "2704399f",
    type: "mcq",
    questionHtml:
      "Which of the following systems of equations has the same solution as the system of equations graphed above?",
    choices: [
      {
        label: "A",
        text: "<strong>Equation 1: y = 0. Equation 2: x = three halves</strong>",
      },
      {
        label: "B",
        text: "<strong>Equation 1: y = three halves. Equation 2: x = 0</strong>",
      },
      {
        label: "C",
        text: "<strong>Equation 1: y = 0. Equation 2: x = 1</strong>",
      },
      {
        label: "D",
        text: "<strong>Equation 1: y = 1. Equation 2: x = 0</strong>",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The solution to a system of equations is the coordinates of the intersection point of the graphs of the equations in the xy-plane. Based on the graph, the solution to the given system of equations is best approximated as <strong>three halves, 0</strong>. In the xy-plane, the graph of <strong>y = 0</strong> is a horizontal line on which every y-coordinate is 0, and the graph of <strong>x = three halves</strong> is a vertical line on which every x-coordinate is <strong>three halves</strong>. These graphs intersect at the point <strong>with coordinates three halves, 0</strong>. Therefore, the system of equations in choice A has the same solution as the given system.Choices B, C, and D are incorrect. If graphed in the xy-plane, these choices would intersect at the points <strong>with coordinates 0, three halves</strong>, <strong>1, 0</strong>, and <strong>0, 1</strong>, respectively, not <strong>three halves, 0</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/2704399f_img1.png",
  },
  {
    id: "2875ba81",
    type: "mcq",
    questionHtml:
      "<strong>6 x + 7 y = 28</strong><br><strong>2 x + 2 y = 10</strong><br>The solution to the given system of equations is <strong>(x, y)</strong>. What is the value of y?",
    choices: [
      { label: "A", text: "<strong>−2</strong>" },
      { label: "B", text: "<strong>7</strong>" },
      { label: "C", text: "<strong>14</strong>" },
      { label: "D", text: "<strong>18</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The given system of linear equations can be solved by the elimination method. Multiplying each side of the second equation in the given system by <strong>3</strong> yields <strong>(2 x + 2 y) (3) = (10) (3)</strong>, or <strong>6 x + 6 y = 30</strong>. Subtracting this equation from the first equation in the given system yields <strong>(6 x + 7 y) − (6 x + 6 y) = (28) − (30)</strong>, which is equivalent to <strong>(6 x − 6 x) + (7 y − 6 y) = 28 − 30</strong>, or <strong>y = −2</strong>. <br>Choice B is incorrect. This is the value of <strong>x</strong>, not the value of <strong>y</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: "5e422ff9",
    type: "mcq",
    questionHtml:
      "In the solution to the system of equations above, what is the value of y ?",
    choices: [
      { label: "A", text: "<strong>−15</strong>" },
      { label: "B", text: "<strong>−9</strong>" },
      { label: "C", text: "<strong>9</strong>" },
      { label: "D", text: "<strong>15</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Multiplying both sides of <strong>y = 2 x − 3</strong> by 5 results in <strong>5 y = 10 x − 15</strong>. Multiplying both sides of <strong>3 y = 5 x</strong> by 2 results in <strong>6 y = 10 x</strong>. Subtracting the resulting equations yields <strong>5 y − 6 y = (10 x − 15, ) − 10 x</strong>, which simplifies to <strong>−y = −15</strong>. Dividing both sides of <strong>−y = −15</strong> by <strong>−1</strong> results in <strong>y = 15</strong>.Choices A and B are incorrect and may result from incorrectly subtracting the transformed equation. Choice C is incorrect and may result from finding the value of x instead of the value of y.",
    hasFigure: false,
  },
  {
    id: "686b7cad",
    type: "mcq",
    questionHtml:
      "A proposal for a new library was included on an election ballot. A radio show stated that <strong>3</strong> times as many people voted in favor of the proposal as people who voted against it. A social media post reported that <strong>15, 000</strong> more people voted in favor of the proposal than voted against it. Based on these data, how many people voted against the proposal?",
    choices: [
      { label: "A", text: "<strong>7, 500</strong>" },
      { label: "B", text: "<strong>15, 000</strong>" },
      { label: "C", text: "<strong>22, 500</strong>" },
      { label: "D", text: "<strong>45, 000</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It's given that a radio show stated that <strong>3</strong> times as many people voted in favor of the proposal as people who voted against it. Let <strong>x</strong> represent the number of people who voted against the proposal. It follows that <strong>3 x</strong> is the number of people who voted in favor of the proposal and <strong>3 x − x</strong>, or <strong>2 x</strong>, is how many more people voted in favor of the proposal than voted against it. It's also given that a social media post reported that <strong>15, 000</strong> more people voted in favor of the proposal than voted against it. Thus, <strong>2 x = 15, 000</strong>. Since <strong>2 x = 15, 000</strong>, the value of <strong>x</strong> must be half of <strong>15, 000</strong>, or <strong>7, 500</strong>. Therefore, <strong>7, 500</strong> people voted against the proposal.<br>Choice B is incorrect. This is how many more people voted in favor of the proposal than voted against it, not the number of people who voted against the proposal.<br>Choice C is incorrect. This is the number of people who voted in favor of the proposal, not the number of people who voted against the proposal.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "6a87902f",
    type: "mcq",
    questionHtml:
      "<strong>y = 2 x + 10</strong><br><strong>y = 2 x − 1</strong><br>At how many points do the graphs of the given equations intersect in the xy-plane?",
    choices: [
      { label: "A", text: "Zero" },
      { label: "B", text: "Exactly one" },
      { label: "C", text: "Exactly two" },
      { label: "D", text: "Infinitely many" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. A system of two linear equations in two variables, <strong>x</strong> and <strong>y</strong>, has zero points of intersection if the lines represented by the equations in the xy-plane are distinct and parallel. The graphs of two lines in the xy-plane represented by equations in slope-intercept form, <strong>y = m x + b</strong>, are distinct if the y-coordinates of their y-intercepts, <strong>b</strong>, are different and are parallel if their slopes, <strong>m</strong>, are the same. For the two equations in the given system, <strong>y = 2 x + 10</strong> and <strong>y = 2 x − 1</strong>, the values of <strong>b</strong> are <strong>10</strong> and <strong>−1</strong>, respectively, and the values of <strong>m</strong> are both <strong>2</strong>. Since the values of <strong>b</strong> are different, the graphs of these lines have different y-coordinates of the y-intercept and are distinct. Since the values of <strong>m</strong> are the same, the graphs of these lines have the same slope and are parallel. Therefore, the graphs of the given equations are lines that intersect at zero points in the xy-plane.<br>Choice B is incorrect. The graphs of a system of two linear equations have exactly one point of intersection if the lines represented by the equations have different slopes. Since the given equations represent lines with the same slope, there is not exactly one intersection point.<br>Choice C is incorrect. The graphs of a system of two linear equations can never have exactly two intersection points.<br>Choice D is incorrect. The graphs of a system of two linear equations have infinitely many intersection points when the lines represented by the equations have the same slope and the same y-coordinate of the y-intercept. Since the given equations represent lines with different y-coordinates of their y-intercepts, there are not infinitely many intersection points.",
    hasFigure: false,
  },
  {
    id: "6e6a3241",
    type: "mcq",
    questionHtml:
      "Which of the following graphs in the xy-plane could be used to solve the system of equations above?",
    choices: [
      {
        label: "A",
        text: "<strong>The answer choice presents the graph of two lines in the xy plane. The numbers − 4 and 4 are indicated on each axis. One line slants upward and to the right. It crosses the x axis at − 5, and crosses the y axis at 1. The other line slants downward and to the right. It crosses the y axis at 4, and crosses the x axis at 2. The two lines intersect above the x axis and to the right of the y axis.</strong>",
      },
      {
        label: "B",
        text: "<strong>The answer choice presents the graph of two lines in the xy plane. The numbers − 4 and 4 are indicated on each axis. One line slants upward and to the right. It crosses the y axis at − 4, and crosses the x axis at 4. The other line slants downward and to the right. It crosses the y axis at 1, and crosses the x axis at 1. The two lines intersect below the x axis and to the right of the y axis.</strong>",
      },
      {
        label: "C",
        text: "<strong>The answer choice presents the graph of two lines in the xy plane. The numbers − 4 and 4 are indicated on each axis. One line slants upward and to the right. It crosses the x axis at − 2, and crosses the y axis at 4. The other line slants downward and to the right. It crosses the y axis at 1, and crosses the x axis at 5. The two lines intersect above the x axis and to the left of the y axis.</strong>",
      },
      {
        label: "D",
        text: "<strong>The answer choice presents the graph of two lines in the xy plane. The numbers − 4 and 4 are indicated on each axis. One line slants upward and to the right. It crosses the x axis at − 2, and crosses the y axis at 4. The other line slants upward and to the right. It crosses the y axis at − 1, and crosses the x axis at 5. The two lines intersect below the x axis and to the left of the y axis.</strong>",
      },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. The graph of a system of equations is the graph that shows the lines represented by each of the equations in the system. The x-intercept of the graph of each given equation can be found by substituting 0 for y in each equation: <strong>x + 5 · 0 = 5</strong>, or <strong>x = 5</strong>, and <strong>2 x − 0 = −4</strong>, or <strong>x = −2</strong>. The y-intercept of the graph of each equation can be found by substituting 0 for x in each equation: <strong>0 + 5 y = 5</strong>, or <strong>y = 1</strong>, and <strong>2 · 0 − y = −4</strong> or <strong>y = 4</strong>. Using these x- and y- intercept values, the line that has equation <strong>x + 5 y = 5</strong> passes through the points <strong>with coordinates 0, 1</strong> and <strong>5, 0</strong>, and the line that has equation <strong>2 x − y = −4</strong> passes through the points <strong>with coordinates 0, 4</strong> and <strong>−2, 0</strong>. Only the lines in choice C pass through these points and can be used to solve the given system of equations.Choices A, B, and D are incorrect. In choices A and B, neither line passes through <strong>0, 1</strong> and <strong>5, 0</strong> or <strong>0, 4</strong> and <strong>−2, 0</strong>. In choice D, although one line passes through <strong>0, 4</strong> and <strong>−2, 0</strong> the other line doesn’t pass through <strong>0, 1</strong> and <strong>5, 0</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/6e6a3241_img2.png",
  },
  {
    id: "71189542",
    type: "mcq",
    questionHtml:
      "A group of 202 people went on an overnight camping trip, taking 60 tents with them. Some of the tents held 2 people each, and the rest held 4 people each. Assuming all the tents were filled to capacity and every person got to sleep in a tent, exactly how many of the tents were 2-person tents?",
    choices: [
      { label: "A", text: "30" },
      { label: "B", text: "20" },
      { label: "C", text: "19" },
      { label: "D", text: "18" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Let x represent the number of 2-person tents and let y represent the number of 4-person tents. It is given that the total number of tents was 60 and the total number of people in the group was 202. This situation can be expressed as a system of two equations, <strong>x + y = 60</strong> and <strong>2 x + 4 y = 202</strong>. The first equation can be rewritten as <strong>y = −x + 60</strong>. Substituting <strong>−x + 60</strong> for y in the equation <strong>2 x + 4 y = 202</strong> yields <strong>2 x + 4 · (−x + 60, ) = 202</strong>. Distributing and combining like terms gives <strong>−2 x + 240 = 202</strong>. Subtracting 240 from both sides of <strong>−2 x + 240 = 202</strong> and then dividing both sides by <strong>−2</strong> gives <strong>x = 19</strong>. Therefore, the number of 2-person tents is 19.Alternate approach: If each of the 60 tents held 4 people, the total number of people that could be accommodated in tents would be 240. However, the actual number of people who slept in tents was 202. The difference of 38 accounts for the 2-person tents. Since each of these tents holds 2 people fewer than a 4-person tent, <strong>thirty eight halves = 19</strong> gives the number of 2-person tents.<br>Choice A is incorrect. This choice may result from assuming exactly half of the tents hold 2 people. If that were true, then the total number of people who slept in tents would be <strong>2 · 30 + 4 · 30 = 180</strong>; however, the total number of people who slept in tents was 202, not 180. Choice B is incorrect. If 20 tents were 2-person tents, then the remaining 40 tents would be 4-person tents. Since all the tents were filled to capacity, the total number of people who slept in tents would be <strong>2 · 20 + 4 · 40 = 40 + 160, which = 200</strong>; however, the total number of people who slept in tents was 202, not 200. Choice D is incorrect. If 18 tents were 2-person tents, then the remaining 42 tents would be 4-person tents. Since all the tents were filled to capacity, the total number of people who slept in tents would be <strong>2 · 18 + 4 · 42 = 36 + 168, which = 204</strong>; however, the total number of people who slept in tents was 202, not 204.",
    hasFigure: false,
  },
  {
    id: "74c03c21",
    type: "mcq",
    questionHtml:
      "A bus traveled on the highway and on local roads to complete a trip of <strong>160 miles</strong>. The trip took <strong>4 hours</strong>. The bus traveled at an average speed of <strong>55 miles per hour (mph)</strong> on the highway and an average speed of <strong>25 mph</strong> on local roads. If <strong>x</strong> is the time, in hours, the bus traveled on the highway and <strong>y</strong> is the time, in hours, it traveled on local roads, which system of equations represents this situation?",
    choices: [
      {
        label: "A",
        text: "<strong>55 x + 25 y = 4</strong><br><strong>x + y = 160</strong>",
      },
      {
        label: "B",
        text: "<strong>55 x + 25 y = 160</strong><br><strong>x + y = 4</strong>",
      },
      {
        label: "C",
        text: "<strong>25 x + 55 y = 4</strong><br><strong>x + y = 160</strong>",
      },
      {
        label: "D",
        text: "<strong>25 x + 55 y = 160</strong><br><strong>x + y = 4</strong>",
      },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. If the bus traveled at an average speed of <strong>55 miles per hour (mph)</strong> on the highway for <strong>x</strong> hours, then the bus traveled <strong>55 x</strong> miles on the highway. If the bus traveled at an average speed of <strong>25 mph</strong> on local roads for <strong>y</strong> hours, then the bus traveled <strong>25 y</strong> miles on local roads. It's given that the trip was <strong>160</strong> miles. This can be represented by the equation <strong>55 x + 25 y = 160</strong>. It's also given that the trip took <strong>4</strong> hours. This can be represented by the equation <strong>x + y = 4</strong>. Therefore, the system consisting of the equations <strong>55 x + 25 y = 160</strong> and <strong>x + y = 4</strong> represents this situation.<br>Choice A is incorrect. This system of equations represents a situation where the trip was <strong>4</strong> miles and took <strong>160</strong> hours.<br>Choice C is incorrect. This system of equations represents a situation where the trip was <strong>4</strong> miles and took <strong>160</strong> hours, and the bus traveled at an average speed of <strong>25 mph</strong> on the highway and <strong>55 mph</strong> on local roads.<br>Choice D is incorrect. This system of equations represents a situation where the bus traveled at an average speed of <strong>25 mph</strong> on the highway and <strong>55 mph</strong> on local roads.",
    hasFigure: false,
  },
  {
    id: "7efe5495",
    type: "mcq",
    questionHtml:
      "<strong>y = 3 x</strong><br><strong>2 x + y = 12</strong><br>The solution to the given system of equations is <strong>(x, y)</strong>. What is the value of <strong>5 x</strong>?",
    choices: [
      { label: "A", text: "<strong>24</strong>" },
      { label: "B", text: "<strong>15</strong>" },
      { label: "C", text: "<strong>12</strong>" },
      { label: "D", text: "<strong>5</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It's given by the first equation in the system that <strong>y = 3 x</strong>. Substituting <strong>3 x</strong> for <strong>y</strong> in the equation <strong>2 x + y = 12</strong> yields <strong>2 x + 3 x = 12</strong>, or <strong>5 x = 12</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 D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "8a87c2c8",
    type: "mcq",
    questionHtml:
      "<strong>x + 3 = − 2 y + 5</strong><br><strong>x − 3 = 2 y + 7</strong><br>The solution to the given system of equations is <strong>(x, y)</strong>. What is the value of <strong>2 x</strong>?",
    choices: [
      { label: "A", text: "<strong>−2</strong>" },
      { label: "B", text: "<strong>6</strong>" },
      { label: "C", text: "<strong>12</strong>" },
      { label: "D", text: "<strong>24</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Adding the second equation in the given system to the first equation in the given system yields <strong>(x + 3) + (x − 3) = (−2 y + 5) + (2 y + 7)</strong>. Adding like terms in this equation yields <strong>2 x = 12</strong>. Thus, the value of <strong>2 x</strong> is <strong>12</strong>.<br>Choice A is incorrect. This is the value of <strong>y</strong>, not <strong>2 x</strong>.<br>Choice B is incorrect. This is the value of <strong>x</strong>, not <strong>2 x</strong>.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "9f6f96ff",
    type: "mcq",
    questionHtml:
      "A wire with a length of <strong>106</strong> inches is cut into two parts. One part has a length of <strong>x</strong> inches, and the other part has a length of <strong>y</strong> inches. The value of <strong>x</strong> is <strong>6</strong> more than <strong>4</strong> times the value of <strong>y</strong>. What is the value of <strong>x</strong>?",
    choices: [
      { label: "A", text: "<strong>25</strong>" },
      { label: "B", text: "<strong>28</strong>" },
      { label: "C", text: "<strong>56</strong>" },
      { label: "D", text: "<strong>86</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It's given that a wire with a length of <strong>106</strong> inches is cut into two parts. It's also given that one part has a length of <strong>x</strong> inches and the other part has a length of <strong>y</strong> inches. This can be represented by the equation <strong>x + y = 106</strong>. It's also given that the value of <strong>x</strong> is <strong>6</strong> more than <strong>4</strong> times the value of <strong>y</strong>. This can be represented by the equation <strong>x = 4 y + 6</strong>. Substituting <strong>4 y + 6</strong> for <strong>x</strong> in the equation <strong>x + y = 106</strong> yields <strong>4 y + 6 + y = 106</strong>, or <strong>5 y + 6 = 106</strong>. Subtracting <strong>6</strong> from each side of this equation yields <strong>5 y = 100</strong>. Dividing each side of this equation by <strong>5</strong> yields <strong>y = 20</strong>. Substituting <strong>20</strong> for <strong>y</strong> in the equation <strong>x = 4 y + 6</strong> yields <strong>x = 4 (20) + 6</strong>, or <strong>x = 86</strong>. <br>Choice A is incorrect. This value represents less than half of the total length of <strong>106</strong> inches; however, <strong>x</strong> represents the length of the longer part of the wire, since it's given that the value of <strong>x</strong> is <strong>6</strong> more than <strong>4</strong> times the value of <strong>y</strong>.<br>Choice B is incorrect. This value represents less than half of the total length of <strong>106</strong> inches; however, <strong>x</strong> represents the length of the longer part of the wire, since it's given that the value of <strong>x</strong> is <strong>6</strong> more than <strong>4</strong> times the value of <strong>y</strong>.<br>Choice C is incorrect. This represents a part that is <strong>6</strong> more than the length of the other part, rather than <strong>6</strong> more than <strong>4</strong> times the length of the other part.",
    hasFigure: false,
  },
  {
    id: "b3c7ca1d",
    type: "mcq",
    questionHtml:
      "Which of the following systems of linear equations has no solution?",
    choices: [
      { label: "A", text: "<strong>x = 3</strong><br><strong>y = 5</strong>" },
      {
        label: "B",
        text: "<strong>y = 6 x + 6</strong><br><strong>y = 5 x + 6</strong>",
      },
      {
        label: "C",
        text: "<strong>y = 16 x + 3</strong><br><strong>y = 16 x + 19</strong>",
      },
      {
        label: "D",
        text: "<strong>y = 5</strong><br><strong>y = 5 x + 5</strong>",
      },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. A system of two linear equations in two variables, <strong>x</strong> and <strong>y</strong>, has no solution if the graphs of the lines represented by the equations in the xy-plane are distinct and parallel. The graphs of two lines in the xy-plane represented by equations in slope-intercept form, <strong>y = m x + b</strong>, where <strong>m</strong> and <strong>b</strong> are constants, are parallel if their slopes, <strong>m</strong>, are the same and are distinct if their y-coordinates of the y-intercepts, <strong>b</strong>, are different. In the equations <strong>y = 16 x + 3</strong> and <strong>y = 16 x + 19</strong>, the values of <strong>m</strong> are each <strong>16</strong>, and the values of <strong>b</strong> are <strong>3</strong> and <strong>19</strong>, respectively. Since the slopes of these lines are the same, and the y-coordinates of the y-intercepts are different, it follows that the system of linear equations in choice C has no solution. <br>Choice A is incorrect. The lines represented by the equations in this system are a vertical line and a horizontal line. Therefore, this system has a solution, <strong>(3, 5)</strong>, rather than no solution.<br>Choice B is incorrect. The two lines represented by these equations have different slopes and the same y-coordinate of the y-intercept. Therefore, this system has a solution, <strong>(0, 6)</strong>, rather than no solution.<br>Choice D is incorrect. The two lines represented by these equations are a horizontal line and a line with a slope of <strong>5</strong> that have the same y-coordinate of the y-intercept. Therefore, this system has a solution, <strong>(0, 5)</strong>, rather than no solution.",
    hasFigure: false,
  },
  {
    id: "b544a348",
    type: "spr",
    questionHtml:
      "In the solution <strong>x, y</strong> to the system of equations above, what is the value of x ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 7. Subtracting the second equation from the first equation eliminates the variable y.<br> <strong>Equation 1, 5 x + 3 y = 38; Equation 2, x + 3 y = 10; Subtracting equation 2 from equation 1 yields equation 3, which is 4 x = 28.</strong><br>Dividing both sides of the resulting equation by 4 yields x = 7.",
    hasFigure: false,
  },
  {
    id: "bf4a8b6a",
    type: "mcq",
    questionHtml:
      "A company that provides whale-watching tours takes groups of <strong>21</strong> people at a time. The company’s revenue is <strong>80</strong> dollars per adult and <strong>60</strong> dollars per child. If the company’s revenue for one group consisting of adults and children was <strong>1, 440</strong> dollars, how many people in the group were children?",
    choices: [
      { label: "A", text: "<strong>3</strong>" },
      { label: "B", text: "<strong>9</strong>" },
      { label: "C", text: "<strong>12</strong>" },
      { label: "D", text: "<strong>18</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Let <strong>x</strong> represent the number of children in a whale-watching tour group. Let <strong>y</strong> represent the number of adults in this group. Because it's given that <strong>21</strong> people are in a group and the group consists of adults and children, it must be true that <strong>x + y = 21</strong>. Since the company's revenue is <strong>60</strong> dollars per child, the total revenue from <strong>x</strong> children in this group was <strong>60 x</strong> dollars. Since the company's revenue is <strong>80</strong> dollars per adult, the total revenue from <strong>y</strong> adults in this group was <strong>80 y</strong> dollars. Because it's given that the total revenue for this group was <strong>1, 440</strong> dollars, it must be true that <strong>60 x + 80 y = 1, 440</strong>. The equations <strong>x + y = 21</strong> and <strong>60 x + 80 y = 1, 440</strong> form a linear system of equations that can be solved to find the value of <strong>x</strong>, which represents the number of children in the group, using the elimination method. Multiplying both sides of the equation <strong>x + y = 21</strong> by <strong>80</strong> yields <strong>80 x + 80 y = 1, 680</strong>. Subtracting <strong>60 x + 80 y = 1, 440</strong> from <strong>80 x + 80 y = 1, 680</strong> yields <strong>(80 x + 80 y) − (60 x + 80 y) = 1, 680 − 1, 440</strong>, which is equivalent to <strong>80 x − 60 x + 80 y − 80 y = 240</strong>, or <strong>20 x = 240</strong>. Dividing both sides of this equation by <strong>20</strong> yields <strong>x = 12</strong>. Therefore, <strong>12</strong> people in the group were children.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect. This is the number of adults in the group, not the number of children in the group.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "c5082ce3",
    type: "spr",
    questionHtml:
      "The score on a trivia game is obtained by subtracting the number of incorrect answers from twice the number of correct answers. If a player answered 40 questions and obtained a score of 50, how many questions did the player answer correctly?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 30. Let x represent the number of correct answers from the player and y represent the number of incorrect answers from the player. Since the player answered 40 questions in total, the equation <strong>x + y = 40</strong> represents this situation. Also, since the score is found by subtracting the number of incorrect answers from twice the number of correct answers and the player received a score of 50, the equation <strong>2 x − y = 50</strong> represents this situation. Adding the equations in the system of two equations together yields <strong>(x + y, ) + (2 x − y, ) = 40 + 50</strong>. This can be rewritten as <strong>3 x = 90</strong>. Finally, solving for x by dividing both sides of the equation by 3 yields <strong>x = 30</strong>.",
    hasFigure: false,
  },
  {
    id: "cb8f449f",
    type: "mcq",
    questionHtml:
      "The system of equations above has solution (x, y). What is the value of x ?",
    choices: [
      { label: "A", text: "<strong>three</strong>" },
      { label: "B", text: "<strong>the fraction 7 over 2</strong>" },
      { label: "C", text: "<strong>four</strong>" },
      { label: "D", text: "<strong>six</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Adding the corresponding sides of the two equations eliminates y and yields <strong>x = 6</strong>, as shown.<strong>The equation one half y = 4, added to the equation x − one half y = 2, gives the equation x + 0 = 6</strong><br>If (x, y) is a solution to the system, then (x, y) satisfies both equations in the system and any equation derived from them. Therefore, <strong>x = 6</strong>.<br>Choices A, B, and C are incorrect and may be the result of errors when solving the system.",
    hasFigure: false,
  },
  {
    id: "d909cd31",
    type: "mcq",
    questionHtml:
      "<strong>− 15 x + 25 y = 65</strong><br>One of the two equations in a system of linear equations is given. The system has infinitely many solutions. Which of the following could be the second equation in the system?",
    choices: [
      { label: "A", text: "<strong>12 x + 20 y = 52</strong>" },
      { label: "B", text: "<strong>12 x + 20 y = −52</strong>" },
      { label: "C", text: "<strong>− 12 x + 20 y = 52</strong>" },
      { label: "D", text: "<strong>− 12 x + 20 y = −52</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It's given that the system has infinitely many solutions. A system of two linear equations has infinitely many solutions when the two linear equations are equivalent. Dividing both sides of the given equation by <strong>5</strong> yields <strong>− 3 x + 5 y = 13</strong>. Dividing both sides of choice C by <strong>4</strong> also yields <strong>− 3 x + 5 y = 13</strong>, so choice C is equivalent to the given equation. Thus, choice C could be the second equation in the system.<br>Choice A is incorrect. The system consisting of this equation and the given equation has one solution, not infinitely many solutions.<br>Choice B is incorrect. The system consisting of this equation and the given equation has one solution, not infinitely many solutions.<br>Choice D is incorrect. The system consisting of this equation and the given equation has no solution, not infinitely many solutions.",
    hasFigure: false,
  },
  {
    id: "e53688cb",
    type: "spr",
    questionHtml:
      "<strong>x + 3 y = 29</strong><br><strong>3 y = 11</strong><br>The solution to the given system of equations is <strong>(x, y)</strong>. What is the value of <strong>x</strong>?",
    choices: [],
    correctAnswer: "18",
    explanation:
      "The correct answer is <strong>18</strong>. It's given by the second equation in the system that <strong>3 y = 11</strong>. Substituting <strong>11</strong> for <strong>3 y</strong> in the first equation in the system, <strong>x + 3 y = 29</strong>, yields <strong>x + 11 = 29</strong>. Subtracting <strong>11</strong> from both sides of this equation yields <strong>x = 18</strong>.",
    hasFigure: false,
  },
  {
    id: "e77a76ce",
    type: "mcq",
    questionHtml:
      "Which of the following systems of linear equations has no solution?",
    choices: [
      {
        label: "A",
        text: "<strong>y = 6 x + 3</strong><br><strong>y = 6 x + 9</strong>",
      },
      {
        label: "B",
        text: "<strong>y = 10</strong><br><strong>y = 10 x + 10</strong>",
      },
      {
        label: "C",
        text: "<strong>y = 14 x + 14</strong><br><strong>y = 10 x + 14</strong>",
      },
      { label: "D", text: "<strong>x = 3</strong><br><strong>y = 10</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. A system of two linear equations in two variables, <strong>x</strong> and <strong>y</strong>, has no solution if the graphs of the lines represented by the equations in the xy-plane are distinct and parallel. The graphs of two lines in the xy-plane represented by equations in slope-intercept form, <strong>y = m x + b</strong>, where <strong>m</strong> and <strong>b</strong> are constants, are parallel if their slopes, <strong>m</strong>, are the same and are distinct if their y-coordinates of the y-intercepts, <strong>b</strong>, are different. In the equations <strong>y = 6 x + 3</strong> and <strong>y = 6 x + 9</strong>, the values of <strong>m</strong> are each <strong>6</strong>, and the values of <strong>b</strong> are <strong>3</strong> and <strong>9</strong>, respectively. Since the slopes of these lines are the same and the y-coordinates of the y-intercepts are different, it follows that the system of linear equations in choice A has no solution.<br>Choice B is incorrect. The two lines represented by these equations are a horizontal line and a line with a slope of <strong>10</strong> that have the same y-coordinate of the y-intercept. Therefore, this system has a solution, <strong>(0, 10)</strong>, rather than no solution.<br>Choice C is incorrect. The two lines represented by these equations have different slopes and the same y-coordinate of the y-intercept. Therefore, this system has a solution, <strong>(0, 14)</strong>, rather than no solution.<br>Choice D is incorrect. The two lines represented by these equations are a vertical line and a horizontal line. Therefore, this system has a solution, <strong>(3, 10)</strong>, rather than no solution.",
    hasFigure: false,
  },
  {
    id: "ed92fb68",
    type: "mcq",
    questionHtml:
      "If the system of equations above has solution <strong>the ordered pair x, y</strong>, what is the value of <strong>x + y</strong> ?",
    choices: [
      { label: "A", text: "0" },
      { label: "B", text: "9" },
      { label: "C", text: "18" },
      { label: "D", text: "38" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Adding the given equations yields 9x + 9y = 162. Dividing each side of the equation 9x + 9y = 162 by 9 gives x + y = 18.Choice A is incorrect and may result from incorrectly adding the equations. Choice B is incorrect and may result from conceptual or computational errors. Choice D is incorrect. This value is equivalent to y – x.",
    hasFigure: false,
  },
  {
    id: "f5929f7a",
    type: "mcq",
    questionHtml:
      "<strong>y = − one ninth x</strong><br><strong>y = one half x</strong><br>The solution to the given system of equations is <strong>(x, y)</strong>. What is the value of <strong>x</strong>?",
    choices: [
      { label: "A", text: "<strong>−9</strong>" },
      { label: "B", text: "<strong>−7</strong>" },
      { label: "C", text: "<strong>0</strong>" },
      { label: "D", text: "<strong>2</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It's given by the first equation in the system that <strong>y = − one ninth x</strong>. Substituting <strong>− one ninth x</strong> for <strong>y</strong> in the second equation in the system yields <strong>− one ninth x = one half x</strong>. Multiplying the left-hand side of this equation by <strong>two halves</strong> and the right-hand side by <strong>nine ninths</strong> yields <strong>− two eighteenths x = nine eighteenths x</strong>. Adding <strong>two eighteenths x</strong> to both sides of this equation yields <strong>0 = (11) / (18) x</strong>. Multiplying both sides of this equation by <strong>(18) / (11)</strong> yields <strong>x = 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 D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
];

export const SYSTEMS_HARD: PracticeQuestion[] = [
  {
    id: "1362ccde",
    type: "spr",
    questionHtml:
      "<strong>y = 4 x + 1</strong><br><strong>4 y = 15 x − 8</strong><br>The solution to the given system of equations is <strong>(x, y)</strong>. What is the value of <strong>x − y</strong>?",
    choices: [],
    correctAnswer: "35",
    explanation:
      "The correct answer is <strong>35</strong>. The first equation in the given system of equations defines <strong>y</strong> as <strong>4 x + 1</strong>. Substituting <strong>4 x + 1</strong> for <strong>y</strong> in the second equation in the given system of equations yields <strong>4 (4 x + 1) = 15 x − 8</strong>. Applying the distributive property on the left-hand side of this equation yields <strong>16 x + 4 = 15 x − 8</strong>. Subtracting <strong>15 x</strong> from each side of this equation yields <strong>x + 4 = −8</strong>. Subtracting <strong>4</strong> from each side of this equation yields <strong>x = −12</strong>. Substituting <strong>−12</strong> for <strong>x</strong> in the first equation of the given system of equations yields <strong>y = 4 (−12) + 1</strong>, or <strong>y = −47</strong>. Substituting <strong>−12</strong> for <strong>x</strong> and <strong>−47</strong> for <strong>y</strong> into the expression <strong>x − y</strong> yields <strong>−12 − (−47)</strong>, or <strong>35</strong>.",
    hasFigure: false,
  },
  {
    id: "14360f84",
    type: "mcq",
    questionHtml:
      "<strong>4 x − 9 y = 9 y + 5</strong><br><strong>h y = 2 + 4 x</strong><br>In the given system of equations, <strong>h</strong> is a constant. If the system has no solution, what is the value of <strong>h</strong>?",
    choices: [
      { label: "A", text: "<strong>−9</strong>" },
      { label: "B", text: "<strong>0</strong>" },
      { label: "C", text: "<strong>9</strong>" },
      { label: "D", text: "<strong>18</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. A system of two linear equations in two variables, <strong>x</strong> and <strong>y</strong>, has no solution if the lines represented by the equations in the xy-plane are distinct and parallel. The graphs of two lines in the xy-plane represented by equations in the form <strong>A x + B y = C</strong>, where <strong>A</strong>, <strong>B</strong>, and <strong>C</strong> are constants, are parallel if the coefficients for <strong>x</strong> and <strong>y</strong> in one equation are proportional to the corresponding coefficients in the other equation. The first equation in the given system can be written in the form <strong>A x + B y = C</strong> by subtracting <strong>9 y</strong> from both sides of the equation to yield <strong>4 x − 18 y = 5</strong>. The second equation in the given system can be written in the form <strong>A x + B y = C</strong> by subtracting <strong>4 x</strong> from both sides of the equation to yield <strong>− 4 x + h y = 2</strong>. The coefficient of <strong>x</strong> in this second equation, <strong>−4</strong>, is <strong>−1</strong> times the coefficient of <strong>x</strong> in the first equation, <strong>4</strong>. For the lines to be parallel, the coefficient of <strong>y</strong> in the second equation, <strong>h</strong>, must also be <strong>−1</strong> times the coefficient of <strong>y</strong> in the first equation, <strong>−18</strong>. Thus, <strong>h = − 1 (−18)</strong>, or <strong>h = 18</strong>. Therefore, if the given system has no solution, the value of <strong>h</strong> is <strong>18</strong>.<br>Choice A is incorrect. If the value of <strong>h</strong> is <strong>−9</strong>, then the given system would have one solution, rather than no solution.<br>Choice B is incorrect. If the value of <strong>h</strong> is <strong>0</strong>, then the given system would have one solution, rather than no solution.<br>Choice C is incorrect. If the value of <strong>h</strong> is <strong>9</strong>, then the given system would have one solution, rather than no solution.",
    hasFigure: false,
  },
  {
    id: "1e11190a",
    type: "mcq",
    questionHtml:
      "Store A sells raspberries for <strong>dollar sign 5.50</strong> per pint and blackberries for <strong>dollar sign 3.00</strong> per pint. Store B sells raspberries for <strong>dollar sign 6.50</strong> per pint and blackberries for <strong>dollar sign 8.00</strong> per pint. A certain purchase of raspberries and blackberries would cost <strong>dollar sign 37.00</strong> at Store A or <strong>dollar sign 66.00</strong> at Store B. How many pints of blackberries are in this purchase?",
    choices: [
      { label: "A", text: "<strong>4</strong>" },
      { label: "B", text: "<strong>5</strong>" },
      { label: "C", text: "<strong>8</strong>" },
      { label: "D", text: "<strong>12</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice C is correct. It’s given that store A sells raspberries for <strong>dollar sign 5 . 50</strong> per pint and blackberries for <strong>dollar sign 3 . 00</strong> per pint, and a certain purchase of raspberries and blackberries at store A would cost <strong>dollar sign 37 . 00</strong>. It’s also given that store B sells raspberries for <strong>dollar sign 6 . 50</strong> per pint and blackberries for <strong>dollar sign 8 . 00</strong> per pint, and this purchase of raspberries and blackberries at store B would cost <strong>dollar sign 66 . 00</strong>. Let <strong>r</strong> represent the number of pints of raspberries and <strong>b</strong> represent the number of pints of blackberries in this purchase. The equation <strong>5 . 50 r + 3 . 00 b = 37 . 00</strong> represents this purchase of raspberries and blackberries from store A and the equation <strong>6 . 50 r + 8 . 00 b = 66 . 00</strong> represents this purchase of raspberries and blackberries from store B. Solving the system of equations by elimination gives the value of <strong>r</strong> and the value of <strong>b</strong> that make the system of equations true. Multiplying both sides of the equation for store A by <strong>6.5</strong> yields <strong>(5 . 50 r) (6 . 5) + (3 . 00 b) (6 . 5) = (37 . 00) (6 . 5)</strong>, or <strong>35 . 75 r + 19 . 5 b = 240 . 5</strong>. Multiplying both sides of the equation for store B by <strong>5.5</strong> yields <strong>(6 . 50 r) (5 . 5) + (8 . 00 b) (5 . 5) = (66 . 00) (5 . 5)</strong>, or <strong>35 . 75 r + 44 b = 363</strong>. Subtracting both sides of the equation for store A, <strong>35 . 75 r + 19 . 5 b = 240 . 5</strong>, from the corresponding sides of the equation for store B, <strong>35 . 75 r + 44 b = 363</strong>, yields <strong>(35 . 75 r − 35 . 75 r) + (44 b − 19 . 5 b) = (363 − 240 . 5)</strong>, or <strong>24 . 5 b = 122 . 5</strong>. Dividing both sides of this equationby <strong>24.5</strong> yields <strong>b = 5</strong>. Thus, <strong>5</strong> pints of blackberries are in<br>this purchase.<br><br>Choices A and B are incorrect and may result from conceptual or calculation errors. Choice D is incorrect. This is the number of pints of raspberries, not blackberries, in the purchase.",
    hasFigure: false,
  },
  {
    id: "27f5fff3",
    type: "mcq",
    questionHtml:
      "For the first line in the system:<br><br>The line slants gradually down from left to right.<br>The line passes through the following points:<br><br>(negative 4 comma 5)<br>(0 comma 4)<br>(8 comma 2)<br><br>For the second line in the system:<br><br>The line is horizontal.<br>The line passes through the following points:<br><br>(0 comma 2)<br>(8 comma 2)<br><br>If a new graph of three linear equations is created using the system of equations shown and the equation <strong>x + 4 y = −16</strong>, how many solutions <strong>(x, y)</strong> will the resulting system of three equations have?",
    choices: [
      { label: "A", text: "Zero" },
      { label: "B", text: "Exactly one" },
      { label: "C", text: "Exactly two" },
      { label: "D", text: "Infinitely many" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. A solution to a system of equations must satisfy each equation in the system. It follows that if an ordered pair <strong>(x, y)</strong> is a solution to the system, the point <strong>(x, y)</strong> lies on the graph in the xy-plane of each equation in the system. The only point that lies on each graph of the system of two linear equations shown is their intersection point <strong>(8, 2)</strong>.  It follows that if a new graph of three linear equations is created using the system of equations shown and the graph of <strong>x + 4 y = −16</strong>, this system has either zero solutions or one solution, the point <strong>(8, 2)</strong>. Substituting <strong>8</strong> for <strong>x</strong> and <strong>2</strong> for <strong>y</strong> in the equation <strong>x + 4 y = −16</strong> yields <strong>8 + 4 (2) = −16</strong>, or <strong>16 = −16</strong>. Since this equation is not true, the point <strong>(8, 2)</strong> does not lie on the graph of <strong>x + 4 y = −16</strong>. Therefore, <strong>(8, 2)</strong> is not a solution to the system of three equations. It follows that there are zero solutions to this system.<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/27f5fff3_svg1.svg",
  },
  {
    id: "45a534d0",
    type: "spr",
    questionHtml:
      "<strong>48 x − 72 y = 30 y + 24</strong><br><strong>r y = one sixth − 16 x</strong><br>In the given system of equations, <strong>r</strong> is a constant. If the system has no solution, what is the value of <strong>r</strong>?",
    choices: [],
    correctAnswer: "-34",
    explanation:
      "The correct answer is <strong>−34</strong>. A system of two linear equations in two variables, <strong>x</strong> and <strong>y</strong>, has no solution if the lines represented by the equations in the xy-plane are distinct and parallel. Two lines represented by equations in standard form <strong>A x + B y = C</strong>, where <strong>A</strong>, <strong>B</strong>, and <strong>C</strong> are constants, are parallel if the coefficients for <strong>x</strong> and <strong>y</strong> in one equation are proportional to the corresponding coefficients in the other equation. The first equation in the given system can be written in standard form by subtracting <strong>30 y</strong> from both sides of the equation to yield <strong>48 x − 102 y = 24</strong>. The second equation in the given system can be written in standard form by adding <strong>16 x</strong> to both sides of the equation to yield <strong>16 x + r y = one sixth</strong>.  The coefficient of <strong>x</strong> in this second equation, <strong>16</strong>, is <strong>one third</strong> times the coefficient of <strong>x</strong> in the first equation, <strong>48</strong>. For the lines to be parallel the coefficient of <strong>y</strong> in the second equation, <strong>r</strong>, must also be <strong>one third</strong> times the coefficient of <strong>y</strong> in the first equation, <strong>−102</strong>. Thus, <strong>r = one third (−102)</strong>, or <strong>r = −34</strong>. Therefore, if the given system has no solution, the value of <strong>r</strong> is <strong>−34</strong>.",
    hasFigure: false,
  },
  {
    id: "466b87e3",
    type: "spr",
    questionHtml:
      "In the system of equations above, c is a constant. If the system has no solution, what is the value of c ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is <strong>one half</strong>. A system of two linear equations has no solution when the graphs of the equations have the same slope and different y-intercepts. Each of the given linear equations is written in the slope-intercept form, <strong>y = m x + b</strong>, where m is the slope and b is the y-coordinate of the y-intercept of the graph of the equation. For these two linear equations, the y-intercepts are <strong>0, 8</strong> and <strong>0, 10</strong>. Thus, if the system of equations has no solution, the slopes of the graphs of the two linear equations must be the same. The slope of the graph of the first linear equation is <strong>one half</strong>. Therefore, for the system of equations to have no solution, the value of c must be <strong>one half</strong>. Note that 1/2 and .5 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "52cb8ea4",
    type: "mcq",
    questionHtml:
      "If <strong>the ordered pair x, y</strong> is the solution to the system of equations above, what is the value of <strong>3 x + 3 y</strong> ?",
    choices: [
      { label: "A", text: "<strong>−13</strong>" },
      { label: "B", text: "<strong>−5</strong>" },
      { label: "C", text: "<strong>5</strong>" },
      { label: "D", text: "<strong>13</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Subtracting the second equation, <strong>4 x − 8 y = 9</strong>, from the first equation, <strong>7 x − 5 y = 4</strong>, results in <strong>(7 x − 5 y, ) − (4 x − 8 y, ) = 4 − 9</strong>, or <strong>7 x − 5 y − 4 x + 8 y = 5</strong>. Combining like terms on the left-hand side of this equation yields <strong>3 x + 3 y = −5</strong>.Choice A is incorrect and may result from miscalculating <strong>4 − 9</strong> as <strong>−13</strong>. Choice C is incorrect and may result from miscalculating <strong>4 − 9</strong> as 5. Choice D is incorrect and may result from adding 9 to 4 instead of subtracting 9 from 4.",
    hasFigure: false,
  },
  {
    id: "567ac7ab",
    type: "mcq",
    questionHtml:
      "One of the two equations in a linear system is <strong>2 x + 6 y = 10</strong>. The system has no solution. Which of the following could be the other equation in the system?",
    choices: [
      { label: "A", text: "<strong>x + 3 y = 5</strong>" },
      { label: "B", text: "<strong>x + 3 y = −20</strong>" },
      { label: "C", text: "<strong>6 x − 2 y = 0</strong>" },
      { label: "D", text: "<strong>6 x + 2 y = 10</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. A system of two linear equations written in standard form has no solution when the equations are distinct and the ratio of the x-coefficient to the y-coefficient for one equation is equivalent to the ratio of the x-coefficient to the y-coefficient for the other equation. This ratio for the given equation is 2 to 6, or 1 to 3. Only choice B is an equation that isn’t equivalent to the given equation and whose ratio of the x-coefficient to the y-coefficient is 1 to 3.Choice A is incorrect. Multiplying each of the terms in this equation by 2 yields an equation that is equivalent to the given equation. This system would have infinitely many solutions. Choices C and D are incorrect. The ratio of the x-coefficient to the y-coefficient in <strong>6 x − 2 y = 0</strong> (choice C) is <strong>−6</strong> to 2, or <strong>−3</strong> to 1. This ratio in <strong>6 x + 2 y = 10</strong> (choice D) is 6 to 2, or 3 to 1. Since neither of these ratios is equivalent to that for the given equation, these systems would have exactly one solution.",
    hasFigure: false,
  },
  {
    id: "5e08a055",
    type: "mcq",
    questionHtml:
      "<strong>y = 6 x + 18</strong><br>One of the equations in a system of two linear equations is given. The system has no solution. Which equation could be the second equation in the system?",
    choices: [
      { label: "A", text: "<strong>− 6 x + y = 18</strong>" },
      { label: "B", text: "<strong>− 6 x + y = 22</strong>" },
      { label: "C", text: "<strong>− 12 x + y = 36</strong>" },
      { label: "D", text: "<strong>− 12 x + y = 18</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. A system of two linear equations in two variables, <strong>x</strong> and <strong>y</strong>, has no solution if the lines represented by the equations in the xy-plane are parallel and distinct. Lines represented by equations in standard form, <strong>A x + B y = C</strong> and <strong>D x + E y = F</strong>, are parallel if the coefficients for <strong>x</strong> and <strong>y</strong> in one equation are proportional to the corresponding coefficients in the other equation, meaning <strong>(D) / (A) = (E) / (B)</strong>; and the lines are distinct if the constants are not proportional, meaning <strong>(F) / (C)</strong> is not equal to <strong>(D) / (A)</strong> or <strong>(E) / (B)</strong>. The given equation, <strong>y = 6 x + 18</strong>, can be written in standard form by subtracting <strong>6 x</strong> from both sides of the equation to yield <strong>− 6 x + y = 18</strong>. Therefore, the given equation can be written in the form <strong>A x + B y = C</strong>, where <strong>A = −6</strong>, <strong>B = 1</strong>, and <strong>C = 18</strong>. The equation in choice B, <strong>− 6 x + y = 22</strong>, is written in the form <strong>D x + E y = F</strong>, where <strong>D = −6</strong>, <strong>E = 1</strong>, and <strong>F = 22</strong>. Therefore,  <strong>(D) / (A) = (−6) / (−6)</strong>, which can be rewritten as <strong>(D) / (A) = 1</strong>; <strong>(E) / (B) = one oneth</strong>, which can be rewritten as <strong>(E) / (B) = 1</strong>; and <strong>(F) / (C) = (22) / (18)</strong>, which can be rewritten as <strong>(F) / (C) = (11) / (9)</strong>. Since <strong>(D) / (A) = 1</strong>, <strong>(E) / (B) = 1</strong>, and <strong>(F) / (C)</strong> is not equal to <strong>1</strong>, it follows that the given equation and the equation <strong>− 6 x + y = 22</strong> are parallel and distinct. Therefore, a system of two linear equations consisting of the given equation and the equation <strong>− 6 x + y = 22</strong> has no solution. Thus, the equation in choice B could be the second equation in the system.<br>Choice A is incorrect. The equation <strong>− 6 x + y = 18</strong> and the given equation represent the same line in the xy-plane. Therefore, a system of these linear equations would have infinitely many solutions, rather than no solution. <br>Choice C is incorrect. The equation <strong>− 12 x + y = 36</strong> and the given equation represent lines in the xy-plane that are distinct and not parallel. Therefore, a system of these linear equations would have exactly one solution, rather than no solution.<br>Choice D is incorrect. The equation <strong>− 12 x + y = 18</strong> and the given equation represent lines in the xy-plane that are distinct and not parallel. Therefore, a system of these linear equations would have exactly one solution, rather than no solution.",
    hasFigure: false,
  },
  {
    id: "70feb725",
    type: "mcq",
    questionHtml:
      "During a month, Morgan ran r miles at 5 miles per hour and biked b miles at 10 miles per hour. She ran and biked a total of 200 miles that month, and she biked for twice as many hours as she ran. What is the total number of miles that Morgan biked during the month?",
    choices: [
      { label: "A", text: "80" },
      { label: "B", text: "100" },
      { label: "C", text: "120" },
      { label: "D", text: "160" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The number of hours Morgan spent running or biking can be calculated by dividing the distance she traveled during that activity by her speed, in miles per hour, for that activity. So the number of hours she ran can be represented by the expression <strong>the fraction r over 5</strong>, and the number of hours she biked can be represented by the expression <strong>the fraction b over 10</strong>. It’s given that she biked for twice as many hours as she ran, so this can be represented by the equation <strong>the fraction b over 10, end fraction = 2 · the fraction r over 5</strong>, which can be rewritten as <strong>b = 4 r</strong>. It’s also given that she ran r miles and biked b miles, and that she ran and biked a total of 200 miles. This can be represented by the equation <strong>r + b = 200</strong>. Substituting <strong>4 r</strong> for b in this equation yields <strong>r + 4 r = 200</strong>, or <strong>5 r = 200</strong>. Solving for r  yields <strong>r = 40</strong>. Determining the number of miles she biked, b, can be found by substituting 40 for r in <strong>r + b = 200</strong>, which yields <strong>40 + b = 200</strong>. Solving for b yields <strong>b = 160</strong>.Choices A, B, and C are incorrect because they don’t satisfy that Morgan biked for twice as many hours as she ran. In choice A, if she biked 80 miles, then she ran 120 miles, which means she biked for 8 hours and ran for 24 hours. In choice B, if she biked 100 miles, then she ran 100 miles, which means she biked for 10 hours and ran for 20 hours. In choice C, if she biked 120 miles, then she ran for 80 miles, which means she biked for 12 hours and ran for 16 hours.",
    hasFigure: false,
  },
  {
    id: "75012ee7",
    type: "mcq",
    questionHtml:
      "<strong>2 x + 3 y = 7</strong><br><strong>10 x + 15 y = 35</strong><br>For each real number <strong>r</strong>, which of the following points lies on the graph of each equation in the xy-plane for the given system?",
    choices: [
      { label: "A", text: "<strong>((r) / (5) + 7 − (r) / (5) + 35)</strong>" },
      { label: "B", text: "<strong>(−(3 r) / (2) + seven halves, r)</strong>" },
      { label: "C", text: "<strong>(r, (2 r) / (3) + seven thirds)</strong>" },
      { label: "D", text: "<strong>(r − (3 r) / (2) + seven halves)</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The two given equations are equivalent because the second equation can be obtained from the first equation by multiplying each side of the equation by <strong>5</strong>. Thus, the graphs of the equations are coincident, so if a point lies on the graph of one of the equations, it also lies on the graph of the other equation. A point <strong>(x, y)</strong> lies on the graph of an equation in the xy-plane if and only if this point represents a solution to the equation. It is sufficient, therefore, to find the point that represents a solution to the first given equation. Substituting the x- and y-coordinates of choice B, <strong>− (3 r) / (2) + seven halves</strong> and <strong>r</strong>, for <strong>x</strong> and <strong>y</strong>, respectively, in the first equation yields <strong>2 (−(3 r) / (2) + seven halves) + 3 r = 7</strong>, which is equivalent to <strong>− 3 r + 7 + 3 r = 7</strong>, or <strong>7 = 7</strong>. Therefore, the point <strong>(−(3 r) / (2) + seven halves, r)</strong> represents a solution to the first equation and thus lies on the graph of each equation in the xy-plane for the given system.<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: "adb0c96c",
    type: "spr",
    questionHtml:
      "<strong>24 x + y = 48</strong><br><strong>6 x + y = 72</strong><br>The solution to the given system of equations is <strong>(x, y)</strong>. What is the value of <strong>y</strong>?",
    choices: [],
    correctAnswer: "80",
    explanation:
      "The correct answer is <strong>80</strong>. Subtracting the second equation in the given system from the first equation yields <strong>(24 x + y) − (6 x + y) = 48 − 72</strong>, which is equivalent to <strong>24 x − 6 x + y − y = −24</strong>, or <strong>18 x = −24</strong>. Dividing each side of this equation by <strong>3</strong> yields <strong>6 x = −8</strong>. Substituting <strong>−8</strong> for <strong>6 x</strong> in the second equation yields <strong>−8 + y = 72</strong>. Adding <strong>8</strong> to both sides of this equation yields <strong>y = 80</strong>. <br>Alternate approach: Multiplying each side of the second equation in the given system by <strong>4</strong> yields <strong>24 x + 4 y = 288</strong>. Subtracting the first equation in the given system from this equation yields <strong>(24 x + 4 y) − (24 x + y) = 288 − 48</strong>, which is equivalent to <strong>24 x − 24 x + 4 y − y = 240</strong>, or <strong>3 y = 240</strong>. Dividing each side of this equation by <strong>3</strong> yields <strong>y = 80</strong>.",
    hasFigure: false,
  },
  {
    id: "d1b66ae6",
    type: "spr",
    questionHtml:
      "If <strong>the ordered pair x, y</strong> satisfies the system of equations above, what is the value of y ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is <strong>three halves</strong>. One method for solving the system of equations for y is to add corresponding sides of the two equations. Adding the left-hand sides gives <strong>(−x + y, ) + (x + 3 y, )</strong>, or 4y. Adding the right-hand sides yields <strong>−3 . 5 + 9 . 5 = 6</strong>. It follows that <strong>4 y = 6</strong>. Finally, dividing both sides of <strong>4 y = 6</strong> by 4 yields <strong>y = six fourths</strong> or <strong>three halves</strong>. Note that 3/2 and 1.5 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "d7bf55e1",
    type: "spr",
    questionHtml:
      "A movie theater sells two types of tickets, adult tickets for $12 and child tickets for $8. If the theater sold 30 tickets for a total of $300, how much, in dollars, was spent on adult tickets? (Disregard the $ sign when gridding your answer.)",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 180. Let a be the number of adult tickets sold and c be the number of child tickets sold. Since the theater sold a total of 30 tickets, a + c = 30. The price per adult ticket is $12, and the price per child ticket is $8. Since the theater received a total of $300 for the 30 tickets sold, it follows that 12a + 8c = 300. To eliminate c, the first equation can be multiplied by 8 and then subtracted from the second equation:<br><br>  <strong>12 a + 8 c = 300, and − 8 a − 8 c = −240, gives 4 a + 0 c = 60</strong>Because the question asks for the amount spent on adult tickets, which is 12a dollars, the resulting equation can be multiplied by 3 to give 3(4a) = 3(60) = 180. Therefore, $180 was spent on adult tickets.<br>Alternate approach: If all the 30 tickets sold were child tickets, their total price would be 30($8) = $240. Since the actual total price of the 30 tickets was $300, the extra $60 indicates that a certain number of adult tickets, a, were sold. Since the price of each adult ticket is $4 more than each child ticket, 4a = 60, and it follows that 12a = 180.",
    hasFigure: false,
  },
  {
    id: "e1248a5c",
    type: "mcq",
    questionHtml:
      "In the system of equations below, a and c are constants.<br> <br> <strong>Equation 1: one half x + one third y = one sixth. Equation 2: a, x + y = c</strong><br> <br>If the system of equations has an infinite number of solutions <strong>x, y</strong>, what is the value of a ?",
    choices: [
      { label: "A", text: "<strong>−one half</strong>" },
      { label: "B", text: "<strong>0</strong>" },
      { label: "C", text: "<strong>one half</strong>" },
      { label: "D", text: "<strong>three halves</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. A system of two linear equations has infinitely many solutions if one equation is equivalent to the other. This means that when the two equations are written in the same form, each coefficient or constant in one equation is equal to the corresponding coefficient or constant in the other equation multiplied by the same number. The equations in the given system of equations are written in the same form, with x and y on the left-hand side and a constant on the right-hand side of the equation. The coefficient of y in the second equation is equal to the coefficient of y in the first equation multiplied by 3. Therefore, a, the coefficient of x in the second equation, must be equal to 3 times the coefficient of x in the first equation: <strong>a = one half · 3</strong>, or <strong>a = three halves</strong>.Choices A, B, and C are incorrect. When <strong>a = −one half</strong>, <strong>a = 0</strong>, or <strong>a = one half</strong>, the given system of equations has one solution.",
    hasFigure: false,
  },
  {
    id: "e2e3942f",
    type: "mcq",
    questionHtml:
      "In the system of equations above, a is a constant. If the system of equations has no solution, what is the value of a ?",
    choices: [
      { label: "A", text: "<strong>−one half</strong>" },
      { label: "B", text: "<strong>0</strong>" },
      { label: "C", text: "<strong>1</strong>" },
      { label: "D", text: "<strong>2</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. A system of two linear equations has no solution when the graphs of the equations have the same slope and different y-coordinates of the y-intercepts. Each of the given equations is written in 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 of the graph of the equation. For these two linear equations, the y-coordinates of the y-intercepts are different: <strong>1</strong> and <strong>−8</strong>. Thus, if the system of equations has no solution, the slopes of the two linear equations must be the same. The slope of the first linear equation is 2. Therefore, for the system of equations to have no solution, the value of a must be 2.Choices A, B, and C are incorrect and may result from conceptual and computational errors.",
    hasFigure: false,
  },
  {
    id: "f03465dc",
    type: "mcq",
    questionHtml:
      "<strong>8 x + 7 y = 9</strong><br><strong>24 x + 21 y = 27</strong><br>For each real number <strong>r</strong>, which of the following points lies on the graph of each equation in the xy-plane for the given system?",
    choices: [
      {
        label: "A",
        text: "<strong>(r − (8 r) / (7) + nine sevenths)</strong>",
      },
      {
        label: "B",
        text: "<strong>(−(8 r) / (7) + nine sevenths, r)</strong>",
      },
      {
        label: "C",
        text: "<strong>(−(8 r) / (7) + 9, (8 r) / (7) + 27)</strong>",
      },
      { label: "D", text: "<strong>((r) / (3) + 9 − (r) / (3) + 27)</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Dividing both sides of the second equation in the given system by <strong>3</strong> yields <strong>8 x + 7 y = 9</strong>, which is the first equation in the given system. Therefore, the first and second equations represent the same line in the xy-plane. If the x- and y-coordinates of a point satisfy an equation, the point lies on the graph of the equation in the xy-plane. Choice A is a point with x-coordinate <strong>r</strong> and y-coordinate <strong>− (8 r) / (7) + nine sevenths</strong>. Substituting <strong>r</strong> for <strong>x</strong> and <strong>− (8 r) / (7) + nine sevenths</strong> for <strong>y</strong> in the equation <strong>8 x + 7 y = 9</strong> yields <strong>8 r + 7 (−eight sevenths r + nine sevenths) = 9</strong>. Applying the distributive property to the left-hand side of this equation yields <strong>8 r − 8 r + 9 = 9</strong>. Combining like terms on the left-hand side of this equation yields <strong>9 = 9</strong>, so the coordinates of the point <strong>(r − eight sevenths r + nine sevenths)</strong> satisfy both equations in the given system. Therefore, for each real number <strong>r</strong>, the point <strong>(r − eight sevenths r + nine sevenths)</strong> lies on the graph of each equation in the xy-plane for the given system.<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: "f718c9cf",
    type: "spr",
    questionHtml:
      "<strong>5 x + 14 y = 45</strong><br><strong>10 x + 7 y = 27</strong><br>The solution to the given system of equations is <strong>(x, y)</strong>. What is the value of <strong>xy</strong>?",
    choices: [],
    correctAnswer: "1.8, 9/5",
    explanation:
      "The correct answer is <strong>nine fifths</strong>. Multiplying the first equation in the given system by <strong>2</strong> yields <strong>10 x + 28 y = 90</strong>. Subtracting the second equation in the given system, <strong>10 x + 7 y = 27</strong>, from <strong>10 x + 28 y = 90</strong> yields <strong>(10 x + 28 y) − (10 x + 7 y) = 90 − 27</strong>, which is equivalent to <strong>10 x + 28 y − 10 x − 7 y = 63</strong>, or <strong>21 y = 63</strong>. Dividing both sides of this equation by <strong>21</strong> yields <strong>y = 3</strong>. The value of <strong>x</strong> can be found by substituting <strong>3</strong> for <strong>y</strong> in either of the two given equations. Substituting <strong>3</strong> for <strong>y</strong> in the equation <strong>10 x + 7 y = 27</strong> yields <strong>10 x + 7 (3) = 27</strong>, or <strong>10 x + 21 = 27</strong>. Subtracting <strong>21</strong> from both sides of this equation yields <strong>10 x = 6</strong>. Dividing both sides of this equation by <strong>10</strong> yields <strong>x = six tenths</strong>, or <strong>x = three fifths</strong>. Therefore, the value of <strong>xy</strong> is <strong>(three fifths) (3)</strong>, or <strong>nine fifths</strong>. Note that 9/5 and 1.8 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "f75bd744",
    type: "spr",
    questionHtml:
      "<strong>4 x − 6 y = 10 y + 2</strong><br><strong>t y = one half + 2 x</strong><br>In the given system of equations, <strong>t</strong> is a constant. If the system has no solution, what is the value of <strong>t</strong>?",
    choices: [],
    correctAnswer: "8",
    explanation:
      "The correct answer is <strong>8</strong>. The given system of equations can be solved using the elimination method. Multiplying both sides of the second equation in the given system by <strong>−2</strong> yields <strong>− 2 t y = −1 − 4 x</strong>, or <strong>−1 − 4 x = − 2 t y</strong>. Adding this equation to the first equation in the given system, <strong>4 x − 6 y = 10 y + 2</strong>, yields <strong>(4 x − 6 y) + (−1 − 4 x) = (10 y + 2) + (−2 t y)</strong>, or <strong>−1 − 6 y = 10 y − 2 t y + 2</strong>. Subtracting <strong>10 y</strong> from both sides of this equation yields <strong>(−1 − 6 y) − (10 y) = (10 y − 2 t y + 2) − (10 y)</strong>, or <strong>−1 − 16 y = − 2 t y + 2</strong>. If the given system has no solution, then the equation <strong>−1 − 16 y = − 2 t y + 2</strong> has no solution. If this equation has no solution, the coefficients of <strong>y</strong> on each side of the equation, <strong>−16</strong> and <strong>− 2 t</strong>, must be equal, which yields the equation <strong>−16 = − 2 t</strong>. Dividing both sides of this equation by <strong>−2</strong> yields <strong>8 = t</strong>. Thus, if the system has no solution, the value of <strong>t</strong> is <strong>8</strong>.<br>Alternate approach: A system of two linear equations in two variables, <strong>x</strong> and <strong>y</strong>, has no solution if the lines represented by the equations in the xy-plane are parallel and distinct. Lines represented by equations in the form <strong>A x + B y = C</strong>, where <strong>A</strong>, <strong>B</strong>, and <strong>C</strong> are constant terms, are parallel if the ratio of the x-coefficients is equal to the ratio of the y-coefficients, and distinct if the ratio of the x-coefficients are not equal to the ratio of the constant terms. Subtracting <strong>10 y</strong> from both sides of the first equation in the given system yields <strong>(4 x − 6 y) − (10 y) = (10 y + 2) − (10 y)</strong>, or <strong>4 x − 16 y = 2</strong>. Subtracting <strong>2 x</strong> from both sides of the second equation in the given system yields <strong>(t y) − (2 x) = (one half + 2 x) − (2 x)</strong>, or <strong>− 2 x + t y = one half</strong>. The ratio of the x-coefficients for these equations is <strong>−two fourths</strong>, or <strong>−one half</strong>. The ratio of the y-coefficients for these equations is <strong>− (t) / (16)</strong>. The ratio of the constant terms for these equations is <strong>(one half) / (2)</strong>, or <strong>one fourth</strong>. Since the ratio of the x-coefficients, <strong>−one half</strong>, is not equal to the ratio of the constants, <strong>one fourth</strong>, the lines represented by the equations are distinct. Setting the ratio of the x-coefficients equal to the ratio of the y-coefficients yields <strong>−one half = − (t) / (16)</strong>. Multiplying both sides of this equation by <strong>−16</strong> yields <strong>(−one half) (−16) = (−(t) / (16)) (−16)</strong>, or <strong>t = 8</strong>. Therefore, when <strong>t = 8</strong>, the lines represented by these equations are parallel. Thus, if the system has no solution, the value of <strong>t</strong> is <strong>8</strong>.",
    hasFigure: false,
  },
  {
    id: "ff501705",
    type: "spr",
    questionHtml:
      "<strong>three halves y − one fourth x = two thirds − three halves y</strong><br><strong>one half x + three halves = p y + nine halves</strong><br>In the given system of equations, <strong>p</strong> is a constant. If the system has no solution, what is the value of <strong>p</strong>?",
    choices: [],
    correctAnswer: "6",
    explanation:
      "The correct answer is <strong>6</strong>. A system of two linear equations in two variables, <strong>x</strong> and <strong>y</strong>, has no solution if the lines represented by the equations in the xy-plane are parallel and distinct. Lines represented by equations in standard form, <strong>A x + B y = C</strong> and <strong>D x + E y = F</strong>, are parallel if the coefficients for <strong>x</strong> and <strong>y</strong> in one equation are proportional to the corresponding coefficients in the other equation, meaning <strong>(D) / (A) = (E) / (B)</strong>; and the lines are distinct if the constants are not proportional, meaning <strong>(F) / (C)</strong> is not equal to <strong>(D) / (A)</strong> or <strong>(E) / (B)</strong>. The first equation in the given system is <strong>three halves y − one fourth x = two thirds − three halves y</strong>. Multiplying each side of this equation by <strong>12</strong> yields <strong>18 y − 3 x = 8 − 18 y</strong>. Adding <strong>18 y</strong> to each side of this equation yields <strong>36 y − 3 x = 8</strong>, or <strong>− 3 x + 36 y = 8</strong>. The second equation in the given system is <strong>one half x + three halves = p y + nine halves</strong>. Multiplying each side of this equation by <strong>2</strong> yields <strong>x + 3 = 2 p y + 9</strong>. Subtracting <strong>2 p y</strong> from each side of this equation yields <strong>x + 3 − 2 p y = 9</strong>. Subtracting <strong>3</strong> from each side of this equation yields <strong>x − 2 p y = 6</strong>. Therefore, the two equations in the given system, written in standard form, are <strong>− 3 x + 36 y = 8</strong> and<strong>x − 2 p y = 6</strong>. As previously stated, if this system has no solution, the lines represented by the equations in the xy-plane are parallel and distinct, meaning the proportion <strong>(1) / (−3) = (−2 p) / (36)</strong>, or <strong>−one third = − (p) / (18)</strong>, is true and the proportion <strong>six eighths = (1) / (−3)</strong> is not true. The proportion <strong>six eighths = (1) / (−3)</strong> is not true. Multiplying each side of the true proportion, <strong>−one third = − (p) / (18)</strong>,  by <strong>−18</strong> yields <strong>6 = p</strong>. Therefore, if the system has no solution, then the value of <strong>p</strong> is <strong>6</strong>.",
    hasFigure: false,
  },
];
