Mathematics • Year 8 • Unit 2 • Lesson 13
Intercepts, Mixed Challenge
Pull together everything from Lesson 13: finding intercepts in standard, gradient-intercept and general form, handling special cases (horizontal, vertical, through the origin). Six mixed problems, one "find the mistake", one open-ended challenge.
1. Mixed problems
Show your working for each. 3 marks each
1.1 Find both intercepts of 4x + 5y = 20.
1.2 Find both intercepts of y = −3x + 9.
1.3 A line has x-intercept (5, 0) and y-intercept (0, 2). What is the equation in the form ax + by = c? (Hint: try 2x + 5y = 10 and check both points satisfy it.)
1.4 Which of the following has an x-intercept of 4? A. y = 2x − 8 B. y = 2x + 4 C. y = x + 4 D. y = −x − 4. Justify your choice by setting y = 0.
1.5 State the intercepts (if any) of the vertical line x = 4 and the horizontal line y = −2. Explain in one sentence each why one of each pair is missing.
1.6 Find both intercepts of y = 2x. (This is the special "through origin" case.) What's unusual about your two intercepts?
2. Find the mistake
A student tried to find both intercepts of 3x + 2y = 12. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why, then re-do correctly. 3 marks
Student's working, intercepts of 3x + 2y = 12:
Line 1: y-intercept: set x = 0 → 3(0) + 2y = 12 → 2y = 12 → y = 6 → (0, 6).
Line 2: x-intercept: set x = 0 → 3(0) + 2y = 12 → y = 6 → (6, 0).
Line 3: So the line passes through (0, 6) and (6, 0).
Line 4: Both intercepts are positive integers, so the line falls from top-left to bottom-right. ✓
(a) Which line contains the mistake?
(b) Explain in one or two sentences what's wrong.
(c) Write out the corrected x-intercept calculation and the corrected x-intercept point.
Stuck? To find an x-intercept set y = 0, not x = 0. The student wrote the wrong variable.3. Open-ended challenge, design three intercept pairs
This question has more than one valid answer. 4 marks
3.1 Find three different straight lines in the form ax + by = c (a, b, c whole numbers) that all share the y-intercept (0, 4) but have different x-intercepts.
For each line you find:
(i) Write down the equation.
(ii) State its x-intercept (compute it; show one line of working).
(iii) Verify your y-intercept is (0, 4).
Bonus: Could you find a line that shares the y-intercept (0, 4) but has NO x-intercept? Describe such a line and explain why.
How did this worksheet feel?
What I'll revisit before next class:
1.1-4x + 5y = 20
y-int (x = 0): 5y = 20, y = 4 → (0, 4). x-int (y = 0): 4x = 20, x = 5 → (5, 0).
1.2, y = −3x + 9
y-int: c = 9 → (0, 9). x-int (y = 0): 0 = −3x + 9 → 3x = 9 → x = 3 → (3, 0).
1.3, Intercepts (5, 0) and (0, 2)
Try 2x + 5y = 10. Check (5, 0): 2(5) + 0 = 10 ✓. Check (0, 2): 0 + 5(2) = 10 ✓. So 2x + 5y = 10 is the equation.
1.4, x-intercept of 4
Set y = 0 in each option. A: 0 = 2x − 8 → x = 4. ✓ B: 0 = 2x + 4 → x = −2. C: 0 = x + 4 → x = −4. D: 0 = −x − 4 → x = −4. Answer: A. y = 2x − 8.
1.5, x = 4 and y = −2
x = 4 (vertical): x-intercept = (4, 0). No y-intercept it never crosses the y-axis because x is fixed at 4.
y = −2 (horizontal): y-intercept = (0, −2). No x-intercept it never crosses the x-axis because y is fixed at −2.
1.6, y = 2x
y-int (x = 0): y = 0 → (0, 0). x-int (y = 0): 0 = 2x → x = 0 → (0, 0). Both intercepts are the same point, the origin. Lines through the origin always have x-intercept = y-intercept = (0, 0).
2, Find the mistake
(a) The mistake is on Line 2.
(b) To find the x-intercept the student should set y = 0, not x = 0. They actually re-did the y-intercept calculation but called the answer the x-intercept and gave (6, 0), neither the working nor the answer matches.
(c) Corrected x-intercept: 3x + 2(0) = 12 → 3x = 12 → x = 4. x-intercept = (4, 0). So the line really passes through (0, 6) and (4, 0).
3, Open-ended challenge (sample solution)
Pick equations of the form ax + y = 4 with different a-values, the y-intercept stays at 4 while the x-intercept changes.
Line 1: x + y = 4. x-int: x = 4 → (4, 0). y-int check: y = 4 → (0, 4) ✓.
Line 2: 2x + y = 4. x-int: 2x = 4 → (2, 0). y-int check: y = 4 → (0, 4) ✓.
Line 3: 4x + y = 4. x-int: 4x = 4 → (1, 0). y-int check: y = 4 → (0, 4) ✓.
Bonus: Yes, the horizontal line y = 4 has y-intercept (0, 4) but never reaches y = 0, so it has no x-intercept. Setting y = 0 gives 4 = 0, which is impossible.
Marking: 1 mark per valid line with both intercepts shown (3 marks). 1 bonus mark for the horizontal-line answer with explanation.