Algebra & Functions · Grades 7–9
How to Solve Linear Equations Step by Step
The golden rule: balance
An equation is a balance. Whatever you do to one side, do to the other. Subtract 2x from both sides to move x terms together; subtract 5 from both sides to move the numbers across.
A full example
3x + 5 = 2x − 7 becomes x = −12, written x = −12. Always check by substituting back.
3x + 5 = 2x − 7
Step-by-step working 4 steps
- 1Write the equation
3x + 5 = 2x − 7
- 2Collect the x terms on the left
3x − 2x = −7 − 5 x = −12
Subtract the smaller x term and the constant from both sides, so x is alone on the left and numbers are on the right.
- 3Divide by the coefficient of x
x = −12 ÷ 1 = −12
- 4Check by substituting back
Left: 3(−12) + 5 = −31 Right: 2(−12) − 7 = −31
Both sides match, so the solution is correct.
No solution
If the x terms cancel and the numbers disagree, no x works: 2x + 3 = 2x + 5 gives “No solution”.
Parallel lines never meet
- The sides can never be equal.
Step-by-step working 3 steps
- 1Write the equation
2x + 3 = 2x + 5
- 2Collect the x terms on the left
2x − 2x = 5 − 3 0x = 2
Subtract the smaller x term and the constant from both sides, so x is alone on the left and numbers are on the right.
- 3Result
0 = 2 is never true, so there is no solution.
Infinitely many solutions
If both sides are identical the equation is always true: 2x + 3 = 2x + 3 gives “Infinitely many solutions”.
Common mistakes
Changing a sign incorrectly
Subtracting a term from both sides is safer than “moving” it.
Dividing only one side
Divide both sides by the coefficient.
Skipping the check
Substitute your answer back into the original equation.
Frequently asked questions
What if x has a fraction coefficient?
Multiply both sides by the denominator, or by the reciprocal of the fraction.
