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 equation of the line with gradient 2 that passes through (3, −1).
Show worked solution
y + 1 = 2(x − 3) = 2x − 6, so y = 2x − 7. Check: 2(3) − 7 = −1 ✓. y = 2x − 5 writes y − 1 instead of y + 1; y = 2x + 5 writes x + 3 instead of x − 3; y = 2x + 7 makes both slips.
2 Find the equation of the line through (−2, 9) and (2, 1). Write it as y = mx + c.
Show worked solution
m = (1 − 9) ÷ (2 − (−2)) = −8 ÷ 4 = −2. Through (2, 1): 1 = −4 + c, so c = 5, and the line is y = −2x + 5. Check (−2, 9): 4 + 5 = 9 ✓.
3 A line with gradient −4 passes through (3, −5). At what value does it cross the y-axis?
Show worked solution
−5 = −4(3) + c = −12 + c, so c = 7. The line is y = −4x + 7, and it crosses the y-axis at 7.
Level 2 · Core
4 The line in the graph passes through P(−2, −1) and Q(4, 3). Which is its equation?
Show worked solution
m = (3 − (−1)) ÷ (4 − (−2)) = 4 ÷ 6 = \(\dfrac{2}{3}\). Through Q: 3 = \(\dfrac{8}{3}\) + c, so c = \(\dfrac{1}{3}\). Multiply y = \(\dfrac{2}{3}\)x + \(\dfrac{1}{3}\) by 3: 3y = 2x + 1, which is 2x − 3y = −1. Check P: −4 + 3 = −1 ✓. 2x − 3y = 1 has the wrong sign on the constant; 2x + 3y = −7 has gradient −\(\dfrac{2}{3}\); 3x − 2y = −4 has gradient \(\dfrac{3}{2}\), run over rise.
5 A line passes through (−1, −2) and (2, 7). Find its gradient and its y-intercept. The point (k, −11) is also on the line: find k.
Show worked solution
m = (7 − (−2)) ÷ (2 − (−1)) = 9 ÷ 3 = 3. 7 = 6 + c, so the y-intercept is 1: y = 3x + 1. Then −11 = 3k + 1, so 3k = −12 and k = −4.
6 Line ℓ passes through A(2, 5) and B(6, −3). At what value of x does ℓ cross the x-axis?
Show worked solution
m = (−3 − 5) ÷ (6 − 2) = −2, and 5 = −4 + c gives c = 9: y = −2x + 9. Put y = 0: 2x = 9, so x = 4.5.
Level 3 · Challenge
7 Three statements:
1 The line through (2, −3) with gradient −1 is x + y = −1.
2 The line through (−1, 4) and (3, 4) is x = 4.
3 The line through (0, 6) and (4, 0) passes through (6, −3).
Which statements are correct?
1 The line through (2, −3) with gradient −1 is x + y = −1.
2 The line through (−1, 4) and (3, 4) is x = 4.
3 The line through (0, 6) and (4, 0) passes through (6, −3).
Which statements are correct?
Show worked solution
y + 3 = −(x − 2) gives y = −x − 1, which is x + y = −1 (1 ✓). Both points in statement 2 have y = 4, so the line is horizontal: y = 4, not x = 4 (2 ✗). m = −6 ÷ 4 = −\(\dfrac{3}{2}\) and c = 6; at x = 6, y = −9 + 6 = −3 (3 ✓). 1 and 3.
8 The line through (−3, 10) and (6, −2) is y = mx + c. Find m as a fraction in simplest form (keep any minus sign on the top), and find c.
Show worked solution
m = −12 ÷ 9 = −4/3. Through (6, −2): −2 = −\(\dfrac{4}{3}\) × 6 + c = −8 + c, so c = 6. Check (−3, 10): 4 + 6 = 10 ✓.
9 A(−2, −7), B(1, −1) and C(6, k) lie on one straight line. Find k.
Show worked solution
m(AB) = (−1 − (−7)) ÷ (1 − (−2)) = 6 ÷ 3 = 2, and −1 = 2 + c gives c = −3: y = 2x − 3. So k = 2(6) − 3 = 9. Check: the gradient of BC is (9 − (−1)) ÷ (6 − 1) = 10 ÷ 5 = 2 ✓.