Year 8 - Linear equations and graphs - Free practice

Year 8 Linear Equations & Graphs Worksheets

Plot, solve, sketch. The core Year 8 algebra skill set.

Quick answer

How do you solve linear equations and graph them in Year 8?

A linear equation is solved by doing the same operation to both sides until the pronumeral is alone. To graph y = mx + c, plot the y-intercept c, then use the gradient m to find more points and draw a straight line. For example, y = 2x + 1 has y-intercept 1 and gradient 2.

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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 8 linear equations and graphs: 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 · Solving a multi-step linear equation
Solve $3x + 7 = 22$ .
Show worked solution
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: $x = 5$
Example 2 · Finding gradient and y-intercept from an equation
Write the equation $2y - 6x = 8$ in the form $y = mx + c$ . State the gradient and $y$ -intercept, then find the coordinates of the point where the line crosses the $x$ -axis.
Show worked solution
Key idea: Rearrange into $y = mx + c$ to read off the gradient $m$ and $y$ -intercept $c$ ; set $y = 0$ to find the $x$ -intercept.
1. Add $6x$ to both sides: $2y = 6x + 8$
2. Divide everything by $2$ : $y = 3x + 4$
3. So the gradient is $m = 3$ and the $y$ -intercept is $c = 4$ , giving the point $(0, 4)$ .
4. Set $y = 0$ to find the $x$ -intercept: $0 = 3x + 4 \implies 3x = -4 \implies x = -\frac{4}{3}$
5. Check: $3\left(-\frac{4}{3}\right) + 4 = -4 + 4 = 0$ ✓
Answer: Gradient $= 3$ , $y$ -intercept $= 4$ , $x$ -intercept at $\left(-\dfrac{4}{3},\, 0\right)$
Example 3 · Linear equations in a real-life context
A plumber charges a $60 call-out fee plus $45 for each hour worked. Mia pays a total of $195. Write and solve a linear equation to find how many hours the plumber worked.
Show worked solution
Key idea: Translate the word problem into a linear equation, then solve using inverse operations.
1. Let $h$ be the number of hours worked. The total cost is: $60 + 45h = 195$
2. Subtract $60$ from both sides: $45h = 135$
3. Divide both sides by $45$ : $h = 3$
4. Check: $60 + 45 \times 3 = 60 + 135 = 195$ ✓
Answer: The plumber worked for $3$ hours.

Common mistakes

Where students usually lose marks

Dividing only part of the equation

Student writes $2y - 6x = 8 \implies y - 6x = 4$ (divided only the right side and $2y$ by $2$ , leaving $6x$ unchanged).

When dividing (or multiplying) to isolate a term, every term on both sides must be divided (or multiplied) by the same value. Rewrite the step as $\frac{2y}{2} - \frac{6x}{2} = \frac{8}{2}$ , giving $y - 3x = 4$ .

Flipping the inequality sign when multiplying or dividing by a positive number

Student solves $3x < 12$ and writes $x > 4$ , flipping the sign even though dividing by a positive number.

The inequality sign only flips when you multiply or divide both sides by a negative number. Dividing $3x < 12$ by the positive number $3$ gives $x < 4$ .

Confusing gradient and y-intercept

For $y = 3x + 4$ , student states gradient $= 4$ and $y$ -intercept $= 3$ .

In $y = mx + c$ , the gradient is always the coefficient of $x$ (the number multiplied by $x$ ), and the $y$ -intercept is the constant term added on. So gradient $= 3$ and $y$ -intercept $= 4$ .

Using a closed circle on a number line for a strict inequality

Student marks $x > 2$ with a closed (filled) circle at $2$ on the number line.

A closed circle means the endpoint is included, which represents $\geq$ or $\leq$ . For a strict inequality such as $x > 2$ , use an open (hollow) circle at $2$ to show that $2$ itself is not part of the solution.

Try one yourself

Your turn

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

Got a different linear equations and graphs question wrong? Upload it to Tuterly and get the worked solution and a fresh one to practise, free.

About this worksheet

Why we built it

Year 8 takes linear equations further: graphing them on the Cartesian plane, solving them algebraically, and starting to work with inequalities. These worksheets mix all three so your student can't just memorise one procedure - they have to recognise what the question is asking.

What's covered

Sub-skills your student will practise

✓Solving multi-step linear equations
✓Graphing y = mx + c style equations on the Cartesian plane
✓Reading the gradient and y-intercept from a graph
✓Solving one-variable linear inequalities
✓Representing inequalities on a number line

From our families

Parents on the practice questions.

“Tuterly has been exactly what we needed since Tom started secondary school. Once the topic tests began, he was struggling to keep up. Now he has unlimited practice, all matched to what his class is actually covering, and his Maths results have improved noticeably.”

Rebecca M.

Year 8 · Maths

“My daughter used to leave revision until the night before. Having a platform she can dip into for fifteen minutes after dinner has completely changed that. She's calmer going into assessments, and honestly, so am I.”

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Year 9 · Maths

“I like that I can just keep doing questions until it clicks. If I get something wrong, it shows me why, and then gives me another one like it.”

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Year 10 · Maths

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Frequently asked

Questions parents ask about linear equations and graphs

What's new in Year 8 linear equations?

Graphing on the Cartesian plane (y = mx + c form), reading gradient and y-intercept from a graph, and solving one-variable inequalities. Year 7 stayed algebraic; Year 8 brings the visual side in.

What does the gradient of a line mean?

How steep the line is - specifically, how much y changes for each unit of x. A gradient of 3 means y goes up 3 for every step right. A negative gradient means the line slopes down.

What's the rule for solving inequalities?

Same as equations, except: if you multiply or divide both sides by a negative number, flip the inequality sign. Forgetting to flip is the single most common Year 8 inequality mistake.

How do students sketch y = mx + c quickly?

Start at the y-intercept (the 'c' value), then use the gradient (the 'm' value) to find a second point - rise over run. Connect them with a straight line.

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