How to Solve Linear Equations

The four-part routine

1. Simplify each side: expand brackets and combine like terms.
2. Collect the unknown on one side and numbers on the other.
3. Isolate by dividing by the coefficient of x.
4. Check by substituting back into the original equation.

Undoing operations in reverse

If x has been multiplied by 4 and then had 9 subtracted, undo it backwards: add 9 first, then divide by 4. This reverse-order thinking is the heart of equation solving.

Special situations

Fractions: multiply both sides by the common denominator. Decimals: multiply by 10 or 100 to clear them, or work directly. Unknown on both sides: subtract the smaller x-term from both sides first. If the unknowns cancel entirely, you have no solution or infinitely many, as explained on the linear equation solver page. For stories that lead to equations, see how to solve word problems.

Worked examples

Example 1: Basic: 4x − 9 = 11

1. Add 9 to both sides: 4x = 20.
2. Divide by 4: x = 5.
3. Check: 4(5) − 9 = 11 ✓

Final answer: x = 5

Example 2: Negative coefficient: 7 − 2x = 15

1. Subtract 7 from both sides: −2x = 8.
2. Divide by −2: x = −4.
3. Check: 7 − 2(−4) = 15 ✓

Final answer: x = −4

Example 3: Unknown on both sides: 6x + 1 = 4x + 9

1. Subtract 4x: 2x + 1 = 9.
2. Subtract 1: 2x = 8.
3. Divide by 2: x = 4.

Final answer: x = 4

Example 4: Fraction: (x + 1)/3 = 5

1. Multiply both sides by 3: x + 1 = 15.
2. Subtract 1.

Final answer: x = 14

Example 5: Decimal: 0.5x + 1.2 = 3.7

1. Subtract 1.2: 0.5x = 2.5.
2. Divide by 0.5.

Final answer: x = 5

Common mistakes to avoid

• Changing only one side of the equation.
• Losing a negative sign when moving a term.
• Dividing only part of one side.
• Stopping before x is isolated.
• Not checking the answer.

Frequently asked questions

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