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Solving Linear Equations

Step-by-step methods for solving equations with one variable.

~12 min Mathematics Aligned with NERDC

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What is a Linear Equation?

A linear equation is an equation where the variable has a power of 1. The goal is to find the value of the variable that makes the equation true.

$$ax + b = c$$

Solving Step by Step

Example: Solve for $x$: $3x + 7 = 22$

Step 1: Subtract 7 from both sides:

$$3x + 7 - 7 = 22 - 7$$ $$3x = 15$$

Step 2: Divide both sides by 3:

$$\frac{3x}{3} = \frac{15}{3}$$ $$x = 5$$

Verification

Always check your answer by substituting back into the original equation:

Check	Calculation	Result
Substitute $x = 5$	$3(5) + 7$	$15 + 7 = 22$ ✓

More Examples

Example 2: Solve for $y$: $2y - 4 = 3y + 1$ $$2y - 4 = 3y + 1$$ $$2y - 3y = 1 + 4$$ $$-y = 5$$ $$y = -5$$

Common Mistake: When you move a term to the other side of the equation, always change its sign. Positive becomes negative, and vice versa.

Calculator Check with JavaScript

function solveLinear(a, b, c) {
    // Solves ax + b = c
    const x = (c - b) / a;
    return x;
}

// 3x + 7 = 22
console.log(solveLinear(3, 7, 22)); // 5

// 2y - 4 = 3y + 1 → -y = 5 → y = -5
console.log(solveLinear(-1, -5, 0)); // 5

Rule of Thumb: Whatever you do to one side of the equation, you must do to the other side. This keeps the equation balanced — like a scale!

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Course contents

Algebraic Expressions

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