Year 10 - Linear equations - Free practice

Year 10 Linear Equations Worksheets & Practice Tests

Multi-step equations, equations from formulas, applied problems.

Quick answer

How do you solve linear equations in Year 10?

Solve linear equations by undoing operations in reverse order: clear fractions by multiplying both sides by the lowest common denominator, expand any brackets, then collect like terms and isolate the pronumeral. For example, 2(x + 3) = 14 gives x + 3 = 7, so x = 4.

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Six questions across Foundation, Standard and Extension, with full worked solutions, written to the Australian Curriculum.

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Worked examples

Year 10 linear equations: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

Example 1 · Multi-step equation with brackets
Solve $3(2x - 4) = 2(x + 6)$ .
Show worked solution
Key idea: Expand the brackets first, then collect like terms and isolate $x$ .
1. Expand: $6x - 12 = 2x + 12$
2. Subtract $2x$ from both sides: $4x - 12 = 12$
3. Add $12$ to both sides: $4x = 24$
4. Divide both sides by $4$ : $x = 6$
Check: $3(2(6)-4) = 3(8) = 24$ and $2(6+6) = 2(12) = 24$ ✓

Answer: $x = 6$
Example 2 · Equation with fractions on both sides
Solve $\dfrac{2x + 1}{3} = \dfrac{x - 2}{4}$ .
Show worked solution
Key idea: Multiply both sides by the lowest common multiple of the denominators to eliminate fractions.
1. The LCM of $3$ and $4$ is $12$ . Multiply both sides by $12$ :
  $4(2x + 1) = 3(x - 2)$
2. Expand: $8x + 4 = 3x - 6$
3. Subtract $3x$ from both sides: $5x + 4 = -6$
4. Subtract $4$ from both sides: $5x = -10$
5. Divide both sides by $5$ : $x = -2$
Check: LHS $= \dfrac{2(-2)+1}{3} = \dfrac{-3}{3} = -1$ and RHS $= \dfrac{-2-2}{4} = \dfrac{-4}{4} = -1$ ✓

Answer: $x = -2$
Example 3 · Solving a problem using a formula
The formula for the perimeter of a rectangle is $P = 2(l + w)$ . A rectangular garden has a perimeter of $38$ metres and a width of $7$ metres. Find the length of the garden.
Show worked solution
Key idea: Substitute the known values into the formula, then solve the resulting linear equation for the unknown.
1. Substitute $P = 38$ and $w = 7$ : $38 = 2(l + 7)$
2. Divide both sides by $2$ : $19 = l + 7$
3. Subtract $7$ from both sides: $l = 12$
Check: $2(12 + 7) = 2(19) = 38$ ✓

Answer: The length of the garden is $12$ metres.

Common mistakes

Where students usually lose marks

Forgetting to expand brackets fully

$3(2x - 4) = 6x - 4$ instead of $6x - 12$

Multiply every term inside the brackets by the number outside. Here, $3 \times (-4) = -12$ , not $-4$ . Write out each multiplication separately if needed.

Multiplying only one side when clearing fractions

$\dfrac{2x+1}{3} = \dfrac{x-2}{4}$ becomes $4(2x+1) = \dfrac{x-2}{4}$ after multiplying only the left side by $12$

Whatever you do to one side of the equation you must do to both sides. Multiply every term on both sides by the LCM.

Sign errors when moving terms across the equals sign

$5x + 4 = -6$ becomes $5x = -6 + 4 = -2$

When you move a term to the other side, change its sign. Subtracting $4$ from both sides gives $5x = -6 - 4 = -10$ , not $-6 + 4$ .

Substituting into a formula without solving for the unknown

Writing $38 = 2(l + 7) = 2l + 7$ and stopping, or not isolating $l$

After substituting, treat the result as a normal linear equation and show clear steps to isolate the unknown variable on one side.

Try one yourself

Your turn

Solve for $x$ : $\dfrac{2x - 3}{5} = 7$

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About this worksheet

Why we built it

Year 10 linear equations are mostly the multi-step variety - fractions on both sides, brackets nested in brackets, and equations derived from rearranged formulas. These worksheets give your student that exact difficulty level so the routine becomes second nature before senior Methods maths.

What's covered

Sub-skills your student will practise

✓Multi-step linear equations with brackets
✓Equations with fractions on both sides
✓Rearranging formulas to make a variable the subject
✓Solving equations derived from worded scenarios
✓Equations leading to fractional or decimal solutions

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Parents on the practice questions.

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Frequently asked

Questions parents ask about linear equations

What's new about Year 10 linear equations?

Multi-step equations with brackets and fractions on both sides, plus equations derived from rearranged formulas (make x the subject of A = B + Cx). Year 10 also covers equations whose solutions involve fractions or decimals.

How do students approach an equation with fractions?

Multiply both sides by the lowest common denominator to clear all fractions in one move. For x/3 + 1/2 = 4, multiply by 6: 2x + 3 = 24, then 2x = 21, x = 10.5.

What does 'make x the subject' mean?

Rearrange the formula so x is alone on one side. For y = 2x + 5, make x the subject: y - 5 = 2x, then x = (y - 5) / 2. Same algebra as solving an equation.

Why are multi-step equations so important for senior Methods maths?

Methods spends a huge fraction of its time rearranging and solving equations within longer problems. Students who plateau at single-step equations stall in Year 11. Fluency built in Year 10 carries through.

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