SAT Topic 1

Linear equations in 1 variable

Solve for x — including fractions, decimals, negatives, and equations with the variable on both sides.

Concept

A linear equation in one variable is anything you can rearrange into ax + b = 0. The job is always the same: isolate x. Use inverse operations and keep both sides equal at every step.

Distribute first to clear parentheses.
Combine like terms on each side.
Move all x-terms to one side, all constants to the other.
Divide by the coefficient of x.

For fractions, multiply through by the LCD to clear them. For decimals, multiply through by 10, 100, … so the coefficients are whole numbers. When you divide or multiply by a negative number, the equation stays an equation — but for inequalities, the inequality flips (we'll cover that in Topic 4).

Worked example 1

Solve `3x + 7 = 22`.

Solution

Step 1. Subtract 7 from both sides: 3x = 15

Step 2. Divide both sides by 3: x = 5

x = 5. Check: 3(5) + 7 = 15 + 7 = 22. ✓

Worked example 2

Solve `(x + 3) / 4 = (x − 1) / 2`.

Solution

Step 1. Multiply both sides by 4 to clear the fractions: x + 3 = 2(x − 1)

Step 2. Distribute on the right: x + 3 = 2x − 2

Step 3. Subtract x: 3 = x − 2

Step 4. Add 2: x = 5

x = 5. Check: (5 + 3)/4 = 2 and (5 − 1)/2 = 2. ✓

Practice test

8 questions on solving linear equations — basics, fractions, decimals, and variables on both sides.

Practice test · 8 questions Question 1 of 8 · Score 0