Mathematics • Year 8 • Unit 2 • Lesson 13

x- and y-Intercepts

Build fluency with the intercept method: set x = 0 for the y-intercept, set y = 0 for the x-intercept. One worked example, one guided fill-in, then eight independent problems.

Build · I Do / We Do / You Do

1. I do — fully worked example

Read every line. The intercept method always uses the same trick: kill one variable by setting it to zero.

Problem. Find the x- and y-intercepts of 2x + 3y = 6.

Step 1 — y-intercept: set x = 0.

2(0) + 3y = 6 → 3y = 6 → y = 2

Reason: the y-axis is the vertical line where x = 0. Substituting kills the x-term and leaves a simple equation in y.

y-intercept: (0, 2)

Step 2 — x-intercept: set y = 0.

2x + 3(0) = 6 → 2x = 6 → x = 3

Reason: the x-axis is the horizontal line where y = 0. Common trap — students often switch these two!

x-intercept: (3, 0)

Answer: x-intercept = (3, 0), y-intercept = (0, 2).

Stuck? Revisit lesson § Card 2/3 — set the OTHER variable to zero. x-intercept → y = 0; y-intercept → x = 0.

2. We do — fill in the missing steps

Find the x- and y-intercepts of 3x + 4y = 12. Fill in each blank. 4 marks

Step 1 — y-intercept: set x = 0.

3( ____ ) + 4y = 12 → 4y = ______ → y = ______

y-intercept: ( ______ , ______ )

Step 2 — x-intercept: set y = 0.

3x + 4( ____ ) = 12 → 3x = ______ → x = ______

x-intercept: ( ______ , ______ )

Sketch check: the line passes through these two intercepts. Both coordinates are positive integers, so the line falls from top-left of the first quadrant to bottom-right. ✓

Stuck? Revisit lesson § Card 4 — "set y = 0 to kill the y-term, leaving an equation only in x".

3. You do — independent practice

Show your working. Foundation (direct intercepts), Standard (rearrange first), Extension (special cases).

Foundation — find both intercepts

3.1 Find the x- and y-intercepts of x + y = 5. 1 mark

3.2 Find the x- and y-intercepts of 2x + y = 8. 1 mark

3.3 Find the x- and y-intercepts of x − y = 4. 1 mark

3.4 Find the y-intercept of y = 2x + 7 (shortcut: read c directly). 1 mark

Standard — gradient-intercept form

3.5 Find both intercepts of y = 2x − 6. (Hint: y-intercept is c. For x-intercept set y = 0.) 2 marks

3.6 Find both intercepts of y = −x + 4 and use them to describe the line. 2 marks

Extension — special cases

3.7 The line y = 3 is horizontal. State its y-intercept and explain why it has NO x-intercept. 2 marks

3.8 Find both intercepts of 5x − 2y = 10. 2 marks

Stuck on 3.7? A horizontal line always has the same y-value. Set y = 0 → 3 = 0 (impossible) — so it never crosses the x-axis.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Section 2 — We do (3x + 4y = 12)

Step 1: 3(0) + 4y = 12 → 4y = 12 → y = 3. y-intercept: (0, 3).
Step 2: 3x + 4(0) = 12 → 3x = 12 → x = 4. x-intercept: (4, 0).

3.1 — x + y = 5

y-int (x = 0): y = 5 → (0, 5). x-int (y = 0): x = 5 → (5, 0).

3.2 — 2x + y = 8

y-int (x = 0): y = 8 → (0, 8). x-int (y = 0): 2x = 8, x = 4 → (4, 0).

3.3 — x − y = 4

y-int (x = 0): −y = 4, y = −4 → (0, −4). x-int (y = 0): x = 4 → (4, 0).

3.4 — y = 2x + 7

y-intercept is c = 7, i.e. the point (0, 7).

3.5 — y = 2x − 6

y-int: c = −6 → (0, −6). x-int: 0 = 2x − 6, 2x = 6, x = 3 → (3, 0).

3.6 — y = −x + 4

y-int: c = 4 → (0, 4). x-int: 0 = −x + 4, x = 4 → (4, 0). Description: a downhill line passing through (0, 4) and (4, 0), with gradient −1.

3.7 — y = 3

Every point on the line has y = 3, so the y-intercept is (0, 3). To find the x-intercept we'd set y = 0, but the line says y = 3 always — so y can never equal 0. The line never crosses the x-axis, so it has no x-intercept.

3.8 — 5x − 2y = 10

y-int (x = 0): −2y = 10, y = −5 → (0, −5). x-int (y = 0): 5x = 10, x = 2 → (2, 0).