What you'll learn
This topic covers solving linear equations — equations where the unknown appears to the first power. In this guide you will learn how to solve one- and two-step equations, equations with brackets, equations with the unknown on both sides, and equations involving fractions. You will also learn how to check your answer. Solving equations is one of the most important algebra skills.
Key terms and definitions
Equation — a statement that two expressions are equal, containing an unknown.
Linear equation — an equation where the unknown is to the power 1 (no x², etc.).
Solve — find the value of the unknown that makes the equation true.
Inverse operation — the operation that undoes another (+ and −, × and ÷).
Balance — keeping both sides equal by doing the same to each side.
Core concepts
The balance method
An equation is like a balance: whatever you do to one side you must do to the other side too. Use inverse operations to undo what is done to the unknown, working in reverse order, until the unknown is by itself. For example, to solve x + 7 = 12, subtract 7 from both sides: x = 5.
Two-step equations
When two operations are applied, undo them in reverse order — deal with addition/subtraction first, then multiplication/division. For 3x + 4 = 19: subtract 4 (3x = 15), then divide by 3 (x = 5).
Equations with brackets
Expand the brackets first, then solve as usual. For 2(x + 3) = 16: expand to 2x + 6 = 16, subtract 6 (2x = 10), divide by 2 (x = 5). Alternatively divide both sides by 2 first.
Unknown on both sides
When the unknown appears on both sides, collect the unknowns on one side (subtract the smaller one) and the numbers on the other. For 5x − 2 = 3x + 8: subtract 3x (2x − 2 = 8), add 2 (2x = 10), divide (x = 5).
Equations with fractions
To clear a fraction, multiply both sides by the denominator. For x/4 = 3, multiply by 4: x = 12. If there are several fractions, multiply through by the lowest common denominator to clear them all at once.
Worked examples
Example 1: Two-step
Solve 4x − 5 = 23.
Add 5: 4x = 28. Divide by 4: x = 7.
Example 2: Brackets
Solve 3(2x − 1) = 21.
Expand: 6x − 3 = 21. Add 3: 6x = 24. Divide: x = 4.
Example 3: Unknown on both sides
Solve 7x + 2 = 4x + 17.
Subtract 4x: 3x + 2 = 17. Subtract 2: 3x = 15. Divide: x = 5.
Common mistakes and how to avoid them
Only changing one side. Always do the same to both sides.
Wrong order of inverses. Undo +/− before ×/÷.
Sign errors with brackets. Multiply every term inside the bracket.
Moving terms without changing sign. Crossing the equals sign flips + to − and vice versa.
Not checking. Substitute your answer back to confirm both sides are equal.
Exam technique for Solving Linear Equations
Use inverse operations to isolate the unknown.
Undo in reverse order — numbers before coefficients.
Expand brackets before collecting terms.
Collect unknowns on one side when they appear on both.
Check by substituting your value back in.
Quick revision summary
A linear equation has the unknown to the power 1, and solving means finding the value that makes it true. Treat the equation as a balance: do the same to both sides, using inverse operations (+/− and ×/÷) to isolate the unknown, undoing operations in reverse order (numbers before coefficients). With brackets, expand first (multiplying every term), then solve. When the unknown is on both sides, collect the unknowns on one side and the numbers on the other before dividing. To clear fractions, multiply both sides by the denominator (or the lowest common denominator if there are several). Always check by substituting your answer back so both sides are equal. The common errors are changing only one side, undoing in the wrong order, sign errors when expanding, and forgetting to change signs across the equals sign. Use inverses, work in reverse order, expand brackets, collect unknowns, and verify your solution.