Solve 4(x - 3) = 2x + 6.

4x - 12 = 2x + 6, 2x = 18, so x = 9

Jess is three times as old as her brother. In 4 years their ages will add to 40. How old is Jess now?

b + 3b + 8 = 40, so b = 8 and Jess is 24

Free worksheet · Australian Curriculum

Linear equations worksheets with answers

Every linear equation is solved the same way: do the same thing to both sides until the pronumeral is on its own. Learn each type, then make a free worksheet with full solutions.

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Linear equations at each year level, matched to the textbook chapters most Victorian schools use: a free printable worksheet with answers, sample questions, and an auto-marked test teachers can share.

Free Year 7 linear equations worksheet

10 questions, then the answers with worked solutions. Printable PDF, no sign-up.

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Year 7 topics covered

Introduction to equations
Solving equations by inspection
Equivalent equations
Solving equations algebraically
Equations with fractions
Equations with brackets

Sample Year 7 linear equations questions

Introduction to equations
Is $x = 4$ a solution of $3x + 5 = 17$ ?
Show the answer
Key idea: A value is a solution if substituting it makes both sides equal.
1. Left side: $3 \times 4 + 5 = 12 + 5 = 17$
2. Right side: $17$
3. Both sides are equal, so $x = 4$ is a solution
Answer: Yes
Solving equations by inspection
Solve by inspection:

(a) $x + 9 = 23$ (b) $7m = 63$ (c) $\dfrac{k}{4} = 6$ (d) $20 - p = 8$
Show the answer
Key idea: Ask: what number makes the equation true?
1. (a) What plus $9$ gives $23$ ? $14$ , since $14 + 9 = 23$
2. (b) $7$ times what is $63$ ? $9$ , since $7 \times 9 = 63$
3. (c) What divided by $4$ is $6$ ? $24$ , since $24 \div 4 = 6$
4. (d) $20$ minus what is $8$ ? $12$ , since $20 - 12 = 8$
Answer: (a) $x = 14$ (b) $m = 9$ (c) $k = 24$ (d) $p = 12$
Equivalent equations
Write the equation that results when you add $5$ to both sides of $x - 5 = 12$ .
Show the answer
Key idea: Doing the same operation to both sides gives an equivalent equation.
1. Left: $x - 5 + 5 = x$
2. Right: $12 + 5 = 17$
3. New equation: $x = 17$
Answer: $x = 17$
Solving equations algebraically
Solve $2x + 5 = 17$ , then verify your solution by substitution.
Show the answer
Key idea: Undo the operations in reverse order, subtract first, then divide.
1. Subtract $5$ from both sides:\
$2x + 5 - 5 = 17 - 5 \implies 2x = 12$
1. Divide both sides by $2$ :\
$x = \frac{12}{2} = 6$
1. Verify by substituting $x = 6$ back into the original equation:\
$2(6) + 5 = 12 + 5 = 17 \checkmark$
Answer: $x = 6$

Free Year 8 linear equations worksheet

10 questions, then the answers with worked solutions. Printable PDF, no sign-up.

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Year 8 topics covered

Reviewing equations
Equivalent equations
Equations with fractions
Equations with the pronumeral on both sides
Equations with brackets

Sample Year 8 linear equations questions

Reviewing equations
Solve $3x + 7 = 22$ .
Show the answer
Key idea: Use inverse operations to isolate $x$ , undoing addition before division.
1. Subtract $7$ from both sides: $3x + 7 - 7 = 22 - 7 \implies 3x = 15$
2. Divide both sides by $3$ : $x = \frac{15}{3} = 5$
3. Check by substituting back: $3(5) + 7 = 15 + 7 = 22$ ✓
Answer: $x = 5$
Equivalent equations
Solve $5x - 8 = 17$ and check your answer.
Show the answer
Key idea: Form equivalent equations by doing the same operation to both sides.
1. Add $8$ to both sides: $5x = 25$
2. Divide both sides by $5$ : $x = 5$
3. Check: $5 \times 5 - 8 = 17$ ✓
Answer: $x = 5$
Equations with fractions
Solve for $x$ : $\dfrac{x + 3}{4} = 5$ .
Show the answer
Key idea: Undo each operation in reverse order to isolate the pronumeral.
1. Multiply both sides by 4: $\dfrac{x+3}{4} \times 4 = 5 \times 4$ , giving $x + 3 = 20$ .
2. Subtract 3 from both sides: $x + 3 - 3 = 20 - 3$ .
3. Simplify: $x = 17$ .
4. Check by substituting back: $\dfrac{17 + 3}{4} = \dfrac{20}{4} = 5$ ✓
Answer: $x = 17$
Equations with the pronumeral on both sides
Solve $7x - 4 = 3x + 20$ .
Show the answer
Key idea: Collect the pronumeral terms on one side by subtracting the smaller one from both sides.
1. Subtract $3x$ from both sides: $4x - 4 = 20$
2. Add $4$ : $4x = 24$ , so $x = 6$
3. Check: $7 \times 6 - 4 = 38$ and $3 \times 6 + 20 = 38$ ✓
Answer: $x = 6$

Free Year 9 linear equations worksheet

10 questions, then the answers with worked solutions. Printable PDF, no sign-up.

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Year 9 topics covered

Solving linear equations with pronumerals on one side
Solving equations with brackets and pronumerals on both sides
Solving word problems

Sample Year 9 linear equations questions

Solving linear equations with pronumerals on one side
Solve:

(a) $5x - 7 = 18$

(b) $\dfrac{x}{4} + 3 = -2$

(c) $12 - 3x = 27$
Show the answer
Key idea: Undo the operations in reverse order, doing the same thing to both sides.
1. (a) Add $7$ : $5x = 25$ ; divide by $5$ : $x = 5$
2. (b) Subtract $3$ : $\frac{x}{4} = -5$ ; multiply by $4$ : $x = -20$
3. (c) Subtract $12$ : $-3x = 15$ ; divide by $-3$ : $x = -5$
Answer: (a) $x = 5$ (b) $x = -20$ (c) $x = -5$
Solving equations with brackets and pronumerals on both sides
Solve $3(2x - 4) = 2x + 8$ .
Show the answer
Key idea: Expand brackets first, then collect like terms and isolate $x$ .
1. Expand the left side: $6x - 12 = 2x + 8$ .
2. Subtract $2x$ from both sides: $4x - 12 = 8$ .
3. Add $12$ to both sides: $4x = 20$ .
4. Divide both sides by $4$ : $x = 5$ .
5. Check: $3(2(5) - 4) = 3(6) = 18$ and $2(5) + 8 = 18$ . ✓
Answer: $x = 5$
Solving word problems
The sum of three consecutive whole numbers is $87$ . Write and solve an equation to find the three numbers.
Show the answer
Key idea: Let the smallest number be $n$ . Consecutive numbers are $n$ , $n + 1$ and $n + 2$ .
1. $n + (n + 1) + (n + 2) = 87$
2. $3n + 3 = 87$ , so $3n = 84$ and $n = 28$
3. The numbers are $28$ , $29$ and $30$ (check: $28 + 29 + 30 = 87$ )
Answer: $28$ , $29$ and $30$
Solving linear equations with pronumerals on one side
Solve $\dfrac{2x + 5}{3} = 7$ .
Show the answer
Key idea: Multiply both sides by the denominator first, then solve the two-step equation.
1. Multiply by $3$ : $2x + 5 = 21$
2. Subtract $5$ : $2x = 16$
3. Divide by $2$ : $x = 8$
Answer: $x = 8$

Free Year 10 linear equations worksheet

10 questions, then the answers with worked solutions. Printable PDF, no sign-up.

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Teachers: share one link and every student is marked straight away, with worked solutions.

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Year 10 topics covered

Solving linear equations

Sample Year 10 linear equations questions

Solving linear equations
Solve $\dfrac{x + 3}{4} = 7$ .
Show the answer
Key idea: Multiply both sides by the denominator to eliminate the fraction, then solve.
1. Multiply both sides by $4$ :\
$\begin{gathered}4 \times \frac{x+3}{4} = 7 \times 4 \\ x + 3 = 28\end{gathered}$
1. Subtract $3$ from both sides:\
$x = 25$
1. Check: $\dfrac{25 + 3}{4} = \dfrac{28}{4} = 7$ ✓
Answer: $x = 25$
Solving linear equations
Solve $\dfrac{x + 1}{3} = \dfrac{2x - 1}{5}$ .
Show the answer
Key idea: Find the LCD of both denominators, multiply every term by it to eliminate all fractions, then solve.
1. The LCD of $3$ and $5$ is $15$ . Multiply both sides by $15$ :\
$\begin{gathered}15 \times \frac{x+1}{3} = 15 \times \frac{2x-1}{5} \\ 5(x + 1) = 3(2x - 1)\end{gathered}$
1. Expand the brackets:\
$5x + 5 = 6x - 3$
1. Subtract $5x$ from both sides:\
$5 = x - 3$
1. Add $3$ to both sides:\
$x = 8$
1. Check: $\dfrac{8+1}{3} = \dfrac{9}{3} = 3$ and $\dfrac{2(8)-1}{5} = \dfrac{15}{5} = 3$ ✓
Answer: $x = 8$
Solving linear equations
Layla and her friend Marcus save up to buy a video game together. Marcus contributes $15 more than Layla. One quarter of Marcus's contribution covers the online activation fee, which costs $12. Let $x$ represent Layla's contribution in dollars. Write an equation and solve it to find how much each person contributes.
Show the answer
Key idea: Translate the real-life information into an equation involving an algebraic fraction, then solve by multiplying both sides by the denominator.
1. Marcus contributes $x + 15$ dollars. One quarter of Marcus's contribution equals $12$ , so:\
$\frac{x + 15}{4} = 12$
1. Multiply both sides by $4$ :\
$x + 15 = 48$
1. Subtract $15$ from both sides:\
$x = 33$
1. Marcus's contribution: $x + 15 = 33 + 15 = 48$ dollars.
2. Check: $\dfrac{48}{4} = 12$ ✓
Answer: Layla contributes $33 and Marcus contributes $48.
Solving linear equations
Solve the equation $\frac{x+1}{3} = \frac{2x-1}{5}$
Show the answer
Key idea: Multiply both sides by the LCD to clear all fractions, then solve the resulting linear equation.
1. Find the LCD of 3 and 5: $\text{LCD} = 15$ .
2. Multiply every term on both sides by 15:\
$15 \times \frac{x+1}{3} = 15 \times \frac{2x-1}{5}$
1. Simplify each side:\
$5(x+1) = 3(2x-1)$
1. Expand the brackets:\
$5x + 5 = 6x - 3$
1. Solve for $x$ :\
$\begin{aligned} 5 + 3 &= 6x - 5x \\ 8 &= x \end{aligned}$
1. Check by substituting $x = 8$ back into the original equation: $\dfrac{8+1}{3} = \dfrac{9}{3} = 3$ and $\dfrac{2(8)-1}{5} = \dfrac{15}{5} = 3$ . Both sides are equal.
Answer: $x = 8$

Free Year 10A linear equations worksheet

10 questions, then the answers with worked solutions. Printable PDF, no sign-up.

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Teachers: share one link and every student is marked straight away, with worked solutions.

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Year 10A topics covered

Solving linear equations
Linear equations involving more complex algebraic fractions10A

Sample Year 10A linear equations questions

Solving linear equations
Solve $\dfrac{x + 3}{4} = 7$ .
Show the answer
Key idea: Multiply both sides by the denominator to eliminate the fraction, then solve.
1. Multiply both sides by $4$ :\
$\begin{gathered}4 \times \frac{x+3}{4} = 7 \times 4 \\ x + 3 = 28\end{gathered}$
1. Subtract $3$ from both sides:\
$x = 25$
1. Check: $\dfrac{25 + 3}{4} = \dfrac{28}{4} = 7$ ✓
Answer: $x = 25$
Linear equations involving more complex algebraic fractions
Solve $\frac{x + 2}{3} - \frac{x - 1}{4} = 2$ .
Show the answer
Key idea: Multiply every term by the lowest common denominator, 12, to clear the fractions.
1. $4(x + 2) - 3(x - 1) = 24$
2. $4x + 8 - 3x + 3 = 24$
3. $x + 11 = 24$ , so $x = 13$
Answer: $x = 13$
Solving linear equations
Solve $\dfrac{x + 1}{3} = \dfrac{2x - 1}{5}$ .
Show the answer
Key idea: Find the LCD of both denominators, multiply every term by it to eliminate all fractions, then solve.
1. The LCD of $3$ and $5$ is $15$ . Multiply both sides by $15$ :\
$\begin{gathered}15 \times \frac{x+1}{3} = 15 \times \frac{2x-1}{5} \\ 5(x + 1) = 3(2x - 1)\end{gathered}$
1. Expand the brackets:\
$5x + 5 = 6x - 3$
1. Subtract $5x$ from both sides:\
$5 = x - 3$
1. Add $3$ to both sides:\
$x = 8$
1. Check: $\dfrac{8+1}{3} = \dfrac{9}{3} = 3$ and $\dfrac{2(8)-1}{5} = \dfrac{15}{5} = 3$ ✓
Answer: $x = 8$
Linear equations involving more complex algebraic fractions
Solve $\frac{x}{5} - 1 = 3$ .
Show the answer
Answer: $x = 20$

Topic guide

Linear equations, explained

From one-step equations in Year 7 to fractions on both sides in Year 10, one type at a time.

1. One-step equations

Use the inverse operation to undo what has been done to the pronumeral:

$x + 7 = 12, x = 5 4x = 28, x = 7 \frac{x}{3} = 6, x = 18$

Adding undoes subtracting, and multiplying undoes dividing.

2. Two-step equations

Undo the adding or subtracting first, then the multiplying or dividing:

$5x - 3 = 22, 5x = 25, x = 5$

Whatever you do to one side, do to the other.

3. Pronumerals on both sides

Move the smaller pronumeral term so the pronumeral is only on one side, then solve as before:

$7x - 4 = 3x + 12$

$4x - 4 = 12, 4x = 16, x = 4$

Subtracting $3x$ from both sides keeps the coefficient positive.

4. Equations with brackets

Expand the brackets first, then collect the pronumerals on one side:

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

$6x - 3 = 4x + 8, 2x = 11, x = 5.5$

Answers do not have to be whole numbers.

5. Equations with fractions

Multiply every term by the lowest common denominator to clear the fractions:

$\frac{x}{2} + \frac{x}{3} = 10 \xrightarrow{\times 6} 3x + 2x = 60, 5x = 60, x = 12$

With one fraction on each side, cross-multiply:

$\frac{2x + 1}{5} = \frac{x - 1}{2}, 2(2x + 1) = 5(x - 1), 4x + 2 = 5x - 5, x = 7$

6. Check, then use it in word problems

Substitute to check: for $x = 4$ in $7x - 4 = 3x + 12$ , the left side is $24$ and the right side is $24$ , so it's right.

Three consecutive whole numbers add to $72$ . Let the smallest be $n$ :

$n + (n + 1) + (n + 2) = 72, 3n + 3 = 72, n = 23$

The numbers are $23$ , $24$ and $25$ .

Quick check: 10 questions

Try each one, then open the answer.

Solve $x - 9 = 4$ .
Answer
$x = 13$
Solve $\frac{x}{5} = 8$ .
Answer
$x = 40$
Solve $3x + 8 = 29$ .
Answer
$x = 7$
Solve $9 - 2x = 15$ .
Answer
$-2x = 6$ , so $x = -3$
Solve $5x + 2 = 2x + 17$ .
Answer
$3x = 15$ , so $x = 5$
Solve $4(x - 3) = 2x + 6$ .
Answer
$4x - 12 = 2x + 6$ , $2x = 18$ , so $x = 9$
Solve $3(x + 4) - 2(x - 1) = 20$ .
Answer
$x + 14 = 20$ , so $x = 6$
Solve $\frac{x}{4} - \frac{x}{6} = 2$ .
Answer
$3x - 2x = 24$ , so $x = 24$
Solve $\frac{x + 3}{2} = \frac{2x - 1}{3}$ .
Answer
$3x + 9 = 4x - 2$ , so $x = 11$
Jess is three times as old as her brother. In $4$ years their ages will add to $40$ . How old is Jess now?
Answer
$b + 3b + 8 = 40$ , so $b = 8$ and Jess is $24$

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These topic pages cover this skill, with worked examples, common mistakes and a question to try.

Year 8 worksheetLinear equations and graphs Year 7 worksheetLinear equations Year 9 worksheetLinear equations and graphs Year 10 worksheetLinear equations Year 10 worksheetLinear equations with algebraic fractions

Questions

What is a linear equation?

A linear equation has a pronumeral raised only to the power of one, such as 3x + 5 = 20. It has one solution, and its graph is a straight line.

How do you solve an equation with pronumerals on both sides?

Subtract the smaller pronumeral term from both sides so the pronumeral appears on one side only, then solve it like a two-step equation. Expand any brackets first.

Do the linear equations worksheets come with answers?

Yes. Every question on a generated worksheet has a fully worked solution, and every example on this page shows each step.

More skill worksheets

Skill worksheetFraction Skill worksheetEquivalent fractions Skill worksheetArea of composite shapes Skill worksheetArea of a triangle Skill worksheetMultiplying fractions Skill worksheetAdding fractions Skill worksheetPercentage Skill worksheetSimultaneous equations

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Choose your year level

Linear equations, explained

1. One-step equations

2. Two-step equations

3. Pronumerals on both sides

4. Equations with brackets

5. Equations with fractions

6. Check, then use it in word problems

Quick check: 10 questions

Generate a free maths worksheet

Practice built from mistakes.

Topic worksheets with worked examples

Questions

What is a linear equation?

How do you solve an equation with pronumerals on both sides?

Do the linear equations worksheets come with answers?

More skill worksheets

Better maths marks start with the next wrong answer.