What is a linear equation?

An equation where the unknown isn't squared or under a root, like 2x + 5 = 17. It has (at most) one solution, and its graph is a straight line, which is where the name comes from.

What's the golden rule for solving equations?

Whatever you do to one side, do to the other, so the equation stays balanced. Undo each operation with its opposite: subtract to undo adding, divide to undo multiplying.

How do I solve an equation with x on both sides?

Move all the x terms to one side and the numbers to the other, by adding or subtracting the same thing on both sides. For 5x + 3 = 2x + 15, subtract 2x from both sides, then subtract 3, to get 3x = 12, so x = 4.

How do I get rid of fractions in an equation?

Multiply every term on both sides by the lowest common denominator. For x/2 + x/3 = 10, multiply by 6 to get 3x + 2x = 60, so x = 12.

What year level learns linear equations?

In the Victorian Curriculum 2.0, solving one-step and two-step equations starts in Year 7, Year 8 adds equations with negatives, brackets and pronumerals on both sides, and Year 10 covers multi-step equations with algebraic fractions. Schools sequence topics differently, so check your course.

How to Solve Linear Equations: Step by Step with Worked Examples (Years 7 to 10)

The balance idea

An equation is like a balanced scale: the left side equals the right side. You can do anything you like to it, as long as you do the same thing to both sides.

To get $x$ on its own, use the inverse (opposite) operation:

To undo	Do this
$+\,5$	subtract 5
$-\,5$	add 5
$\times\, 5$	divide by 5
$\div\, 5$	multiply by 5

One-step equations

Worked examples

$x + 7 = 12$ : subtract 7 from both sides.

x = 5

$x - 4 = 9$ : add 4 to both sides.

x = 13

$3x = 21$ : divide both sides by 3.

x = 7

$\dfrac{x}{5} = 4$ : multiply both sides by 5.

x = 20

Two-step equations

Undo the operations in reverse order. In $2x + 5$ , $x$ was multiplied by 2 and then 5 was added. So subtract 5 first, then divide by 2.

Worked example

Solve $2x + 5 = 17$ .

2x = 12 \quad (\text{subtract 5 from both sides})

x = 6 \quad (\text{divide both sides by 2})

More worked examples

$\dfrac{x}{3} - 2 = 4$ : add 2, then multiply by 3.

\frac{x}{3} = 6

x = 18

$5 - 2x = 11$ : subtract 5, then divide by $-2$ .

-2x = 6

x = -3

Equations with x on both sides

Get all the $x$ terms on one side first. It's usually easiest to move the smaller $x$ term, so you don't end up with a negative.

Worked example

Solve $5x + 3 = 2x + 15$ .

3x + 3 = 15 \quad (\text{subtract } 2x)

3x = 12 \quad (\text{subtract 3})

x = 4

Worked example with negatives

Solve $3x - 8 = 5x + 2$ . The smaller $x$ term is $3x$ , so subtract it.

-8 = 2x + 2

-10 = 2x

x = -5

Equations with brackets

Expand the brackets first, then solve as normal.

Worked example

Solve $3(x - 4) = 15$ .

3x - 12 = 15

3x = 27

x = 9

You could also divide both sides by 3 first: $x - 4 = 5$ , so $x = 9$ .

Worked example: brackets on both sides

Solve $2(x + 5) = 4(x - 1)$ .

2x + 10 = 4x - 4

10 = 2x - 4 \quad (\text{subtract } 2x)

14 = 2x

x = 7

Equations with fractions

Clear the fractions by multiplying every term on both sides by the denominator (or the lowest common denominator if there's more than one).

Worked example

Solve $\dfrac{x + 3}{4} = 5$ .

x + 3 = 20 \quad (\text{multiply both sides by 4})

x = 17

Worked example: two fractions

Solve $\dfrac{x}{2} + \dfrac{x}{3} = 10$ . The lowest common denominator is 6.

3x + 2x = 60 \quad (\text{multiply every term by 6})

5x = 60

x = 12

Worked example: a fraction on each side (Year 10)

Solve $\dfrac{2x - 1}{3} = \dfrac{x + 4}{2}$ . Multiply both sides by 6 (or cross-multiply).

2(2x - 1) = 3(x + 4)

4x - 2 = 3x + 12

x = 14

Setting up word problems

Choose a letter for the unknown and say what it stands for.
Turn the words into an equation.
Solve it.
Answer the question in words, with units.

Worked example

I think of a number, multiply it by 3 and add 7. The answer is 31. What's my number?

Let the number be $n$ .

3n + 7 = 31

3n = 24

n = 8

The number is 8.

Worked example

A taxi charges $4 to start plus $2.50 per kilometre. A trip cost $29. How far was it?

Let the distance be $d$ km.

4 + 2.5d = 29

2.5d = 25

d = 10

The trip was 10 km.

Worked example

A rectangle is 3 cm longer than it is wide. Its perimeter is 34 cm. Find its length and width.

Let the width be $w$ cm, so the length is $w + 3$ cm.

2w + 2(w + 3) = 34

4w + 6 = 34

4w = 28

w = 7

The rectangle is 7 cm wide and 10 cm long.

Checking your answer

Substitute your answer into the original equation. If both sides come out the same, it's right.

For $3x - 8 = 5x + 2$ with $x = -5$ :

Left side: $3 \times (-5) - 8 = -23$
Right side: $5 \times (-5) + 2 = -23$

Both sides are $-23$ , so $x = -5$ is correct.

Common mistakes: only doing an operation to one side, forgetting to multiply every term when clearing fractions, and dropping the negative when dividing by a negative number.

Practice questions with answers

Try each one before you open the answer.

Solve $x - 8 = 15$ .
Show the answer
Add 8 to both sides: $x = 23$
Solve $4x + 9 = 33$ .
Show the answer
$4x = 24$ , so $x = 6$
Solve $7 - 3x = 22$ .
Show the answer
$-3x = 15$ , so $x = -5$
Solve $6x - 5 = 2x + 11$ .
Show the answer
$4x - 5 = 11$ , so $4x = 16$ and $x = 4$
Solve $4(x + 3) = 2(x - 1)$ .
Show the answer
$4x + 12 = 2x - 2$ , so $2x = -14$ and $x = -7$
Solve $\dfrac{x - 2}{5} = 3$ .
Show the answer
Multiply by 5: $x - 2 = 15$ , so $x = 17$
Solve $\dfrac{x}{4} + \dfrac{x}{6} = 5$ .
Show the answer
Multiply every term by 12: $3x + 2x = 60$ , so $5x = 60$ and $x = 12$
Three consecutive whole numbers add to 72. Find them.
Show the answer
Let them be $n$ , $n + 1$ and $n + 2$ . Then $3n + 3 = 72$ , so $3n = 69$ and $n = 23$ . The numbers are 23, 24 and 25.

Linear equations by year level

Following the Victorian Curriculum 2.0. Schools sequence topics differently, so check your course. Once you can solve one equation, the next step is two at once: simultaneous equations (Year 10).

Year 7

Linear equations

One-step and two-step equations, using inverse operations and checking by substitution.

Worksheets and topic test · Free PDF with answers

Year 8

Linear equations and graphs

Equations with negatives, brackets and x on both sides, and linear graphs.

Worksheets and topic test · Free PDF with answers

Year 10

Linear equations

Multi-step equations, including rearranging formulas.

Worksheets and topic test · Free PDF with answers

Year 10

Linear equations with algebraic fractions

Equations with algebraic fractions, cleared with the lowest common denominator.

Worksheets and topic test · Free PDF with answers

How to solve linear equations

The balance idea

One-step equations

Two-step equations

Equations with x on both sides

Equations with brackets

Equations with fractions

Setting up word problems

Checking your answer

Practice questions with answers

Linear equations by year level

Solving linear equations: common questions

More maths help, Years 7 to 10

Got an equation wrong in homework?