How to Solve Linear Equations on the SAT

What is a Linear Equation?

A linear equation is any equation that can be written in the form:

$ax + b = c$

where $a$, $b$, and $c$ are constants, and $x$ is the variable. The key feature: the variable has an exponent of 1 (no $x^2$, no $\sqrt{x}$).

How to Solve Linear Equations

Step 1: Simplify Both Sides

Distribute and combine like terms on each side separately.

$3(x + 2) - 4 = 2x + 8$

$3x + 6 - 4 = 2x + 8$

$3x + 2 = 2x + 8$

Step 2: Move Variable Terms to One Side

Subtract $2x$ from both sides:

$3x - 2x + 2 = 8$

$x + 2 = 8$

Step 3: Isolate the Variable

Subtract 2 from both sides:

$x = 6$

Step 4: Verify Your Answer

Plug $x = 6$ back into the original equation:

$3(6 + 2) - 4 = 2(6) + 8$

$3(8) - 4 = 12 + 8$

$24 - 4 = 20 \checkmark$

SAT-Specific Strategies

Always check your answer by substituting back. This takes 5 seconds and catches sign errors, which are the #1 mistake on SAT linear equations.

What the SAT Tests

The SAT rarely gives you a simple "solve for x" problem. Instead, it wraps linear equations in:

No Solution vs. Infinite Solutions

This is a high-frequency SAT question typeAppears on almost every SAT:

• No solution: Same $x$ coefficient, different constants → $3x + 5 = 3x + 8$
• Infinite solutions: Same $x$ coefficient, same constants → $3x + 5 = 3x + 5$
• One solution: Different $x$ coefficients → $3x + 5 = 2x + 8$

Common Mistakes

1. Sign errors when moving terms — When you move $+3$ to the other side, it becomes $-3$, not $+3$
2. Forgetting to distribute — $2(x + 3) = 2x + 6$, not $2x + 3$
3. Dividing only one side — If you divide by 2, divide the entire side, not just one term

Practice Flow

graph TD
    A[Read the problem] --> B{Is it a word problem?}
    B -->|Yes| C[Define variable, write equation]
    B -->|No| D[Simplify both sides]
    C --> D
    D --> E[Move variables to one side]
    E --> F[Isolate x]
    F --> G[Check your answer]
    G -->|Correct| H[Done!]
    G -->|Wrong| D

Key Formulas