How to solve linear equations step by step
An equation is a balance. Whatever you do to one side, you must do to the other. Solving it means performing the same operations on both sides until the unknown stands alone.
Isolating the unknown
Undo the operations in reverse order: first additions and subtractions, then multiplications and divisions.
Example: 3x + 5 = 20
- Subtract 5 from both sides: 3x = 15
- Divide both sides by 3: x = 5
x = 5
Unknowns on both sides
Example: 5x − 4 = 2x + 11
- Subtract 2x: 3x − 4 = 11
- Add 4: 3x = 15
- Divide by 3: x = 5
x = 5
Brackets and fractions
Expand brackets first, or divide both sides by the number in front of them if that is simpler. To remove a fraction, multiply both sides by its denominator.
Example: 2(x − 3) = 10
- Divide by 2: x − 3 = 5
- Add 3: x = 8
x = 8
Example: x/4 + 1 = 6
- Subtract 1: x/4 = 5
- Multiply by 4: x = 20
x = 20
Always check
Replace x with your answer in the original equation. For 3x + 5 = 20 with x = 5, we get 15 + 5 = 20, which is correct. This takes a few seconds and catches most sign errors.
Common pitfalls
- Changing only one side of the equation.
- Multiplying only the first term when expanding: 2(x − 3) is 2x − 6, not 2x − 3.
- Losing a minus sign when moving a term from one side to the other.
Worked exercises
Try to find the answer first, then open the solution.
Solve 7x − 2 = 33.
Show the solution
- 7x = 35
- x = 5
Answer: x = 5
Solve 4(x + 1) = 2x + 10.
Show the solution
- 4x + 4 = 2x + 10
- 2x = 6
- x = 3
Answer: x = 3
Solve x/3 − 2 = 5.
Show the solution
- x/3 = 7
- x = 21
Answer: x = 21