import React, { useRef, useState, useEffect } from "react"; import { ArrowDown, Check, BookOpen, Scale, ArrowRight, RotateCcw, } from "lucide-react"; import LinearSolutionsWidget from "../../../components/lessons/LinearSolutionsWidget"; import Quiz from "../../../components/lessons/Quiz"; import { LINEAR_EQ_QUIZ_DATA } from "../../../utils/constants"; import { Frac } from "../../../components/Math"; interface LessonProps { onFinish?: () => void; } const BalanceScaleWidget = () => { const [left, setLeft] = useState(15); const [right, setRight] = useState(15); const [tilt, setTilt] = useState(0); const [message, setMessage] = useState("Balanced"); const apply = (val: number, side: "both" | "left" | "right") => { let newLeft = left; let newRight = right; if (side === "left" || side === "both") newLeft += val; if (side === "right" || side === "both") newRight += val; setLeft(newLeft); setRight(newRight); if (newLeft === newRight) { setTilt(0); setMessage("Perfectly Balanced! ✅"); } else if (newLeft > newRight) { setTilt(-15); setMessage("Unbalanced! Left side is heavier. ❌"); } else { setTilt(15); setMessage("Unbalanced! Right side is heavier. ❌"); } }; const reset = () => { setLeft(15); setRight(15); setTilt(0); setMessage("Balanced"); }; return (
Goal: Keep the scale balanced while isolating the variable.
Linear equations are like a balance scale. Whatever you do to one side, you must do to the other. The goal is always to isolate the variable — get x by itself on one side with a number on the other.
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Example 1: Basic (no fractions)
Solve: 3(2x − 4) = 2x + 8
Step 1 (Distribute): 6x − 12 = 2x + 8
Step 3 (Move): 4x = 20
Step 4 (Divide by 4):{" "} x = 5
Example 2: With fractions — clear them first!
Solve:
Multiply every term by 6 (LCD of 3 and 2):
2x + 12 = 3x − 6
12 + 6 = 3x − 2x
x = 18
Example 3: Literal equations (isolating a variable)
Solve for r: A = P(1 + rt)
Divide both sides by P:
Subtract 1:
Divide by t:{" "}
r ={" "}
Common SAT Mistake: Distributing a Negative
When distributing a negative sign: −3(x − 4) = −3x + 12, NOT −3x − 12. The negative multiplies EVERY term inside the parentheses.
Not all equations have exactly one answer. After simplifying, look at what's left. The SAT frequently asks for a value of k{" "} that makes an equation have no solution or infinitely many solutions — this is a critical concept to master.
Example: For what value of k does 4x + k = 4x − 2 have no solution?
After subtracting 4x from both sides: k = −2
If k = −2: −2 = −2 (always true) → infinite solutions, not no solution!
For no solution: k ≠ −2 (any other value gives a false statement like k = −2 being false)
The equation has no solution for any value of k EXCEPT −2. For infinite solutions, set k = −2.
Example: For what value of k does 3(x + k) = 3x + 12 have infinite solutions?
Distribute: 3x + 3k = 3x + 12
Subtract 3x: 3k = 12
k = 4
For k = 4: 12 = 12 (always true) → infinite solutions. ✓
The Key Rule
No solution: same variable coefficients,
different constants (parallel lines).
Infinite solutions: same variable coefficients
AND same constants (identical lines).