Solving Linear Equations

Q: What is a linear equation?

A linear equation has a variable raised only to the first power, with no squares or higher powers, so its graph is a straight line.

Q: Why do I do the same thing to both sides?

An equation is a balance. Whatever you do to one side you must do to the other to keep both sides equal.

What is a linear equation?

A linear equation is an equation where the variable (usually x) appears only to the first power — no x², no square roots. Examples include x + 4 = 9 and 3x − 2 = 10.

"Solving" an equation means finding the value of the variable that makes both sides equal. If x + 4 = 9, the solution is x = 5, because 5 + 4 really does equal 9.

If you are new to working with letters in maths, start with algebra basics, then come back here.

The balance rule

Picture an equation as a balance scale. The = sign means both sides weigh the same. To keep it balanced, whatever you do to one side you must do to the other.

This is the single most important idea in solving equations:

Add, subtract, multiply or divide — just do it to both sides.

Inverse operations

To get the variable on its own, you undo the operations around it using their inverses:

Operation	Inverse
Addition (+)	Subtraction (−)
Subtraction (−)	Addition (+)
Multiplication (×)	Division (÷)
Division (÷)	Multiplication (×)

One-step equations

Solve x + 7 = 12. The variable has 7 added to it, so subtract 7 from both sides:

x + 7 = 12
x + 7 − 7 = 12 − 7
x = 5

Solve 3x = 21. The variable is multiplied by 3, so divide both sides by 3:

3x = 21
3x ÷ 3 = 21 ÷ 3
x = 7

Two-step equations

When there is more than one operation, undo addition and subtraction first, then multiplication and division.

Solve 2x + 5 = 17.

2x + 5 = 17
2x = 17 − 5      (subtract 5 from both sides)
2x = 12
x = 12 ÷ 2       (divide both sides by 2)
x = 6

Check: 2 × 6 + 5 = 12 + 5 = 17. Correct.

Variables on both sides

When the variable appears on both sides, collect the variable terms on one side and the numbers on the other.

Solve 5x − 3 = 2x + 9.

5x − 3 = 2x + 9
5x − 2x − 3 = 9       (subtract 2x from both sides)
3x − 3 = 9
3x = 9 + 3            (add 3 to both sides)
3x = 12
x = 4                 (divide both sides by 3)

Check: left side 5 × 4 − 3 = 17; right side 2 × 4 + 9 = 17. Both equal, so x = 4.

Equations with brackets

Expand the brackets first, then solve as usual.

Solve 3(x + 2) = 18.

3(x + 2) = 18
3x + 6 = 18          (multiply out the bracket)
3x = 12              (subtract 6)
x = 4                (divide by 3)

Equations with fractions

Multiply every term by the denominator to clear the fraction.

Solve x/4 = 5. Multiply both sides by 4:

x/4 × 4 = 5 × 4
x = 20

A reliable step-by-step method

Remove brackets and fractions if there are any.
Collect variable terms on one side, numbers on the other.
Use inverse operations to isolate the variable.
Divide to get a single x.
Check by substituting your answer back in.

Master these steps and you can solve any linear equation. To see where these skills lead, revisit algebra basics and look ahead to graphing straight lines.

Key takeaways