Solving Linear Equations Step by Step

Worked example · Mathematics · 1 min read

A clear method for solving equations with one unknown, with worked examples and checks.

A linear equation has a variable raised only to the power one. The method is to do the same operation to both sides until the variable is alone.

The method

1. Expand brackets if there are any.
2. Collect variable terms on one side and numbers on the other.
3. Simplify.
4. Divide by the coefficient of the variable.
5. Check by substituting.

Example 1: two steps

Solve 3x + 5 = 20.

• Subtract 5 from both sides: 3x = 15.
• Divide both sides by 3: x = 5.
• Check: 3(5) + 5 = 20. ✓

Example 2: variable on both sides

Solve 5x − 4 = 2x + 11.

• Subtract 2x from both sides: 3x − 4 = 11.
• Add 4 to both sides: 3x = 15.
• Divide by 3: x = 5.
• Check: left 5(5) − 4 = 21, right 2(5) + 11 = 21. ✓

Example 3: with brackets

Solve 2(x + 3) = 14.

• Expand: 2x + 6 = 14.
• Subtract 6: 2x = 8.
• x = 4.
• Check: 2(4 + 3) = 14. ✓

Example 4: with fractions

Solve x/3 + 2 = 6.

• Subtract 2: x/3 = 4.
• Multiply by 3: x = 12.
• Check: 12/3 + 2 = 6. ✓

Common mistakes

• Changing the sign wrongly when moving a term. Instead of "moving", think "subtract the same thing from both sides".
• Forgetting to multiply every term when clearing a fraction.
• Skipping the check, which catches most arithmetic slips.

Quick recap

Expand, collect, simplify, divide, check.