Year 9 Linear relationships: free practice test with answers

Sample linear relationships questions with answers

Try each one before you open the answer.

1. Introduction to linear relationships

   For the rule $y = 5 - 2x$, find $y$ when:

   (a) $x = 3$

   (b) $x = -4$

   (c) $x = 0.5$

   Show the answer

   Key idea: Substitute each value of $x$ into the rule.

   1. $5 - 2(3) = -1$
   2. $5 - 2(-4) = 13$
   3. $5 - 2(0.5) = 4$

   Answer: (a) $-1$ (b) $13$ (c) $4$

2. Graphing straight lines using intercepts

   Find the $x$-intercept and the $y$-intercept of the line $4x - 3y = 24$.

   Show the answer

   Key idea: At the $x$-intercept $y = 0$, and at the $y$-intercept $x = 0$.

   1. $x$-intercept: $4x - 0 = 24$, so $x = 6$
   2. $y$-intercept: $0 - 3y = 24$, so $y = -8$

   Answer: $x$-intercept $6$, $y$-intercept $-8$

3. Lines with one intercept

   Describe each line and state its one intercept.

   (a) $y = 4$ (b) $x = -2$ (c) $y = -3x$

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   Key idea: $y = c$ is a horizontal line, $x = k$ is a vertical line, and $y = mx$ passes through the origin.

   1. (a) Every point has $y = 4$, so the line is horizontal and crosses the $y$-axis at $4$
   2. (b) Every point has $x = -2$, so the line is vertical and crosses the $x$-axis at $-2$
   3. (c) When $x = 0$, $y = 0$, so the line passes through $(0, 0)$, and this is both intercepts

   Answer: (a) horizontal line, $y$-intercept $4$ (b) vertical line, $x$-intercept $-2$ (c) a line through the origin, intercept $(0, 0)$

4. Gradient and direct proportion

   A tap fills a tank at a constant rate. After 3 hours the tank holds 480 litres. Write an equation in the form $y = kx$ relating the volume of water $y$ (in litres) to the time $x$ (in hours), and find the volume after 7 hours.

   Show the answer

   Key idea: In a direct proportion relationship $y = kx$, the constant $k$ is found by dividing a known $y$-value by the matching $x$-value.

   1. Find $k$: $k = \frac{y}{x} = \frac{480}{3} = 160$
   2. Write the equation: $y = 160x$
   3. Substitute $x = 7$: $y = 160 \times 7 = 1120$

   Answer: The equation is $y = 160x$ and the volume after 7 hours is 1120 litres.

5. Gradient–intercept form

   Sketch the line $y = 2x - 3$ by identifying the gradient and $y$-intercept.

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   Key idea: In $y = mx + c$, $m$ is the gradient and $c$ is the $y$-intercept. Plot the $y$-intercept, then use the gradient to find a second point.

   1. Identify $m = 2$ and $c = -3$, so the $y$-intercept is $(0, -3)$.
   2. From $(0, -3)$, apply the gradient: rise $= 2$, run $= 1$, giving the second point $(1, -1)$.
   3. Plot $(0, -3)$ and $(1, -1)$ and draw a straight line through them.

   Answer: A straight line passing through $(0, -3)$ and $(1, -1)$.

6. Finding the equation of a line

   Find the equation of the line with gradient $-2$ that passes through the point $(3, 1)$.

   Show the answer

   Key idea: Substitute the gradient and the point into $y = mx + c$ to find $c$.

   1. $y = -2x + c$
   2. $1 = -2(3) + c$, so $1 = -6 + c$ and $c = 7$
   3. $y = -2x + 7$

   Answer: $y = -2x + 7$

7. Parallel and perpendicular lines

   Line $\ell_1$ passes through $P(-2, 1)$ and $Q(4, 4)$. Line $\ell_2$ passes through $R(0, 5)$ and $S(6, -7)$. Determine whether $\ell_1$ and $\ell_2$ are parallel, perpendicular, or neither.

   Show the answer

   Key idea: Two lines are parallel if their gradients are equal, and perpendicular if the product of their gradients is $-1$.

   1. Gradient of $\ell_1$: $m_1 = \frac{4 - 1}{4 - (-2)} = \frac{3}{6} = \frac{1}{2}$
   2. Gradient of $\ell_2$: $m_2 = \frac{-7 - 5}{6 - 0} = \frac{-12}{6} = -2$
   3. Check parallel: $m_1 \neq m_2$, so not parallel.
   4. Check perpendicular: $m_1 \times m_2 = \frac{1}{2} \times (-2) = -1 \checkmark$

   Answer: $\ell_1$ and $\ell_2$ are perpendicular.

8. Linear modelling

   A candle is $30$ cm tall. It burns down at a steady rate of $2.5$ cm per hour.

   (a) Write a rule for the height, $h$ cm, after $t$ hours.

   (b) Find the height after $5$ hours.

   (c) How long does the candle take to burn down completely?

   Show the answer

   Key idea: The starting value is the vertical intercept and the rate of change is the gradient. Burning down is a decrease, so the gradient is negative.

   1. (a) $h = 30 - 2.5t$
   2. (b) $h = 30 - 2.5 \times 5 = 30 - 12.5 = 17.5$
   3. (c) $0 = 30 - 2.5t$, so $2.5t = 30$ and $t = 12$

   Answer: (a) $h = 30 - 2.5t$ (b) $17.5$ cm (c) $12$ hours

Free worksheets for this chapter

Worked examples and a free PDF with answers for each topic.

• Year 9 linear equations and graphs worksheet
• Year 9 coordinate geometry worksheet
• Year 9 direct proportion, rates and scale worksheet