Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
Level 1 · Warm-up
1 Find the gradient and the y-intercept of the line 2y − 6x = 10.
Show worked solution
2y = 6x + 10, so y = 3x + 5: m = 3, c = 5. m = −3 forgets that −6x becomes +6x when it changes sides; c = 10 divides only the x term by 2; m = 6, c = 10 reads the numbers off before dividing by 2.
2 Find the x-intercept and the y-intercept of the line 3x − 4y = 24.
Show worked solution
y = 0: 3x = 24, so the x-intercept is 8. x = 0: −4y = 24, so the y-intercept is −6.
3 What is the gradient of the line 6x + 3y = 7?
Show worked solution
3y = −6x + 7, so y = −2x + \(\dfrac{7}{3}\) and the gradient is −2. The 7 only changes the y-intercept.
Level 2 · Core
4 Which equation matches the line in the graph?
Show worked solution
The y-intercept is −3. From (0, −3) to (2, 0) is across 2 and up 3, so m = \(\dfrac{3}{2}\): y = \(\dfrac{3}{2}\)x − 3. y = \(\dfrac{2}{3}\)x − 3 is run over rise (inverted); y = −\(\dfrac{3}{2}\)x − 3 has the wrong sign, though the line rises; y = \(\dfrac{3}{2}\)x + 2 uses the x-intercept 2 as c.
5 The line y = \(\dfrac{4}{5}\)x + c passes through (10, 1). Find c, and then the x-intercept of the line.
Show worked solution
Substitute (10, 1): 1 = \(\dfrac{4}{5}\) × 10 + c = 8 + c, so c = −7. x-intercept: 0 = \(\dfrac{4}{5}\)x − 7, so \(\dfrac{4}{5}\)x = 7 and x = \(\dfrac{35}{4}\) = 8.75 (35/4 is also accepted).
6 The line kx + 2y = 12 has x-intercept −3. What is its gradient?
Show worked solution
At (−3, 0): −3k = 12, so k = −4. Then −4x + 2y = 12, so 2y = 4x + 12 and y = 2x + 6. The gradient is 2.
Level 3 · Challenge
7 Three statements:
1 The line 2x − 3y = 9 has y-intercept 3.
2 y = −\(\dfrac{2}{5}\)x + 4 and 2x + 5y = 20 are the same line.
3 The line through (0, −2) with gradient 5 has x-intercept 0.4.
Which statements are correct?
1 The line 2x − 3y = 9 has y-intercept 3.
2 y = −\(\dfrac{2}{5}\)x + 4 and 2x + 5y = 20 are the same line.
3 The line through (0, −2) with gradient 5 has x-intercept 0.4.
Which statements are correct?
Show worked solution
x = 0: −3y = 9, so the y-intercept is −3, not 3 (1 ✗). 5y = −2x + 20 gives y = −\(\dfrac{2}{5}\)x + 4, the same line (2 ✓). The line is y = 5x − 2; y = 0 gives x = \(\dfrac{2}{5}\) = 0.4 (3 ✓). 2 and 3.
8 The line below meets the x-axis at A and the y-axis at B. Find the midpoint of AB.
Show worked solution
y = 0: 3x = 18, so A is (6, 0). x = 0: −2y = 18, so B is (0, −9). Midpoint: ((6 + 0) ÷ 2, (0 + (−9)) ÷ 2) = (3, −4.5).
9 The line y = −2x + c, with c > 0, makes a triangle of area 36 with the two axes. Find c.
Show worked solution
The y-intercept is c. Put y = 0: 2x = c, so the x-intercept is c ÷ 2. Area = ½ × c × c ÷ 2 = c2 ÷ 4 = 36, so c2 = 144 and c = 12, since c > 0. Check: intercepts 12 and 6, and ½ × 12 × 6 = 36 ✓.