How to Solve Linear Equations Step by Step

Linear equations are the building blocks of algebra. Whether you're in middle school or preparing for the SAT, mastering them gives you the foundation for everything from quadratic equations to calculus. This guide walks you through every type of linear equation with clear, step-by-step solutions.

What Is a Linear Equation?

A linear equation is any equation where the variable has an exponent of 1. The general form is:

ax + b = c

Where:

a = coefficient of x (cannot be 0)

b = constant on the left side

c = constant on the right side

Solving One-Variable Linear Equations

Step-by-Step Method

Goal: Isolate the variable on one side.

Example: Solve $3x + 7 = 22$

Step	Action	Result
1	Subtract 7 from both sides	$3x = 15$
2	Divide both sides by 3	$x = 5$
3	Verify: $3(5) + 7 = 22$

Equations with Variables on Both Sides

Example: Solve $5x - 3 = 2x + 9$

Move variables to one side:

5x - 2x = 9 + 3

2. Simplify: $3x = 12$

3. Divide: $x = 4$

Equations with Fractions

Example: Solve $\frac{x}{3} + \frac{x}{4} = 7$

Find LCD = 12

2. Multiply every term by 12: $4x + 3x = 84$

3. Combine: $7x = 84$

4. Solve: $x = 12$

Systems of Linear Equations

When you have two unknowns, you need two equations:

\begin{cases} 2x + y = 10 \\ x - y = 2 \end{cases}

Method 1: Substitution

From equation 2: $x = y + 2$ . Substitute into equation 1:

$2(y + 2) + y = 10 → 3y = 6 → y = 2, x = 4$

Method 2: Elimination

Add both equations: $(2x + y) + (x - y) = 12 → 3x = 12 → x = 4$

Common Mistakes to Avoid

Forgetting to apply operations to BOTH sides — if you subtract 3 from the left, you must subtract 3 from the right

2. Sign errors — when moving terms across the equals sign, flip the sign

3. Dividing by the coefficient BEFORE simplifying — always combine like terms first

4. Forgetting to check your answer — always plug it back in!

Word Problems with Linear Equations

Problem: "A store sells notebooks for $3 each and pens for$ 1.50 each. If Sam bought 12 items totaling $27, how many of each did he buy?"

Let $n$ = notebooks, $p$ = pens:

n + p = 12

3n + 1.5p = 27

Using substitution: $p = 12 - n$ , so $3n + 1.5(12 - n) = 27$ :

$3n + 18 - 1.5n = 27 → 1.5n = 9 → n = 6$ , $p = 6$

Practice Problems

Try these yourself, then check with AhaStep:

4x - 9 = 2x + 7

2. $\frac{2x+1}{3} = 5$

3. $\begin{cases} 3x + 2y = 16 \\ x - y = 1 \end{cases}$

4. "A train travels at 60 km/h. A car leaves the same station 1 hour later at 80 km/h. When does the car catch the train?"

Why Use AI to Learn Equations?

AhaStep's AI doesn't just give you the answer — it shows you why each step works:

📐 Step-by-step breakdown with explanations

🔍 Method identification — substitution vs. elimination vs. graphing

✅ Verification at the end

🎯 Guide-Only mode — get hints without spoilers

Try it free at ahastep.com

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How to Solve Linear Equations Step by Step — With AI Tutor