How to Solve Linear Equations Step-by-Step

What Is a Linear Equation?

A linear equation is an equation where the variable (usually x) has a power of 1 — no squares, cubes, or other powers. The graph of a linear equation is always a straight line.

Examples:

• 2x + 3 = 7
• x - 5 = 12
• 3x + 2 = 5x - 8

The goal when solving a linear equation is to isolate the variable (get x by itself on one side of the equation).

The Golden Rule of Equations

The most important rule in algebra:

Whatever you do to one side of the equation, you must do to the other side.

This keeps the equation balanced, like a scale. If both sides are equal and you add 5 to both sides, they're still equal.

Operations you can perform on both sides:

• Add or subtract a number
• Multiply or divide by a number (except dividing by zero)

One-Step Equations

These are the simplest — they require only one operation to solve.

Example 1: x + 7 = 12 Subtract 7 from both sides: x + 7 - 7 = 12 - 7 x = 5 ✓

Example 2: x - 3 = 10 Add 3 to both sides: x - 3 + 3 = 10 + 3 x = 13 ✓

Example 3: 4x = 20 Divide both sides by 4: 4x/4 = 20/4 x = 5 ✓

Example 4: x/3 = 6 Multiply both sides by 3: (x/3) × 3 = 6 × 3 x = 18 ✓

Two-Step Equations

These require two operations. The general strategy: undo addition/subtraction first, then undo multiplication/division.

Example: 2x + 5 = 13

Step 1: Subtract 5 from both sides: 2x + 5 - 5 = 13 - 5 2x = 8

Step 2: Divide both sides by 2: 2x/2 = 8/2 x = 4 ✓

Check: 2(4) + 5 = 8 + 5 = 13 ✓

Example: 3x - 7 = 14

Step 1: Add 7 to both sides: 3x = 21

Step 2: Divide by 3: x = 7 ✓

Multi-Step Equations

These involve additional steps like distributing or combining like terms.

Example: 3(x + 2) - 4 = 11

Step 1: Distribute the 3: 3x + 6 - 4 = 11

Step 2: Combine like terms on the left: 3x + 2 = 11

Step 3: Subtract 2: 3x = 9

Step 4: Divide by 3: x = 3 ✓

Check: 3(3 + 2) - 4 = 3(5) - 4 = 15 - 4 = 11 ✓

Example with fractions: (x + 4)/2 = 5

Step 1: Multiply both sides by 2: x + 4 = 10

Step 2: Subtract 4: x = 6 ✓

Variables on Both Sides

When x appears on both sides, move all variable terms to one side and all constant terms to the other.

Example: 5x + 3 = 2x + 15

Step 1: Subtract 2x from both sides: 3x + 3 = 15

Step 2: Subtract 3 from both sides: 3x = 12

Step 3: Divide by 3: x = 4 ✓

Check: 5(4) + 3 = 23 and 2(4) + 15 = 23 ✓

Example: 7x - 2 = 4x + 10

Subtract 4x: 3x - 2 = 10 Add 2: 3x = 12 Divide by 3: x = 4 ✓

Special Cases

Not every linear equation has exactly one solution:

No solution (contradiction): 2x + 3 = 2x + 7 Subtract 2x: 3 = 7 This is false! There is no value of x that makes this true.

Infinite solutions (identity): 2x + 4 = 2(x + 2) 2x + 4 = 2x + 4 This is always true! Every value of x is a solution.

Tips and Tricks

1. Always check your answer: Plug your solution back into the original equation
2. Deal with fractions early: Multiply both sides by the LCD (least common denominator) to eliminate fractions
3. Be careful with negatives: When multiplying or dividing by a negative, be extra cautious with signs
4. Show your work: Write each step clearly — this helps you catch mistakes and earn partial credit on exams
5. Keep the equation balanced: Whatever you do to one side, do to the other

Summary

Solving linear equations is about isolating the variable through inverse operations. Start simple (one-step), build to multi-step, and always check your work. Remember the golden rule: what you do to one side, you must do to the other. With practice, these steps become second nature.