Draw the graph of the following equations, and identify their slopes and y-intercepts. Also, find the coordinates of the points where these lines cut the y-axis.
- (i) y = −3x + 4
- (ii) 2y = 4x + 7
- (iii) 5y = 6x − 10
- (iv) 3y = 6x − 11
- Parallel lines: Are any of the lines parallel?
Step-by-step solution
Idea: Rewrite each equation as y = ax + b by dividing by the coefficient of y. Then a is the slope, b is the y-intercept, and the line cuts the y-axis at (0, b).
(i) y = −3x + 4
- Already in the form y = ax + b: slope −3, y-intercept 4, cuts the y-axis at (0, 4).½ mark
- Points to plot: (0, 4) and (1, 1). The line falls from left to right.½ mark
(ii) 2y = 4x + 7
- Divide by 2: y = 2x + 72. Slope 2, y-intercept 72 (3.5), cuts the y-axis at (0, 72).½ mark
- Points to plot: (0, 3.5) and (1, 5.5).½ mark
(iii) 5y = 6x − 10
- Divide by 5: y = 65x − 2. Slope 65, y-intercept −2, cuts the y-axis at (0, −2).½ mark
- Points to plot: (0, −2) and (5, 4).½ mark
(iv) 3y = 6x − 11
- Divide by 3: y = 2x − 113. Slope 2, y-intercept −113 (about −3.67), cuts the y-axis at (0, −113).½ mark
- Points to plot: (0, −113) and (4, 133), or (1, −53).½ mark
Parallel lines: Are any of the lines parallel?
- Lines with equal slopes are parallel. (ii) and (iv) both have slope 2 (and different y-intercepts), so lines (ii) and (iv) are parallel. No other pair has equal slopes (−3, 2, 65, 2).1 mark
Check: (iii): (5, 4) satisfies 5y = 6x − 10, since 20 = 30 − 10 ✓. (iv): (4, 133) gives 3 × 133 = 13 = 24 − 11 ✓.
Answer to write in the exam
(i)
y = −3x + 4
Slope = −3, y-intercept = 4
Points (0, 4), (1, 1); graph drawn
∴ Cuts the y-axis at (0, 4)
(ii)
y = 2x + 72
Slope = 2, y-intercept = 72
Points (0, 3.5), (1, 5.5); graph drawn
∴ Cuts the y-axis at (0, 72)
(iii)
y = 65x − 2
Slope = 65, y-intercept = −2
Points (0, −2), (5, 4); graph drawn
∴ Cuts the y-axis at (0, −2)
(iv)
y = 2x − 113
Slope = 2, y-intercept = −113
Points (0, −113), (1, −53); graph drawn
∴ Cuts the y-axis at (0, −113)
Parallel lines:
Slopes: −3, 2, 65, 2
(ii) and (iv): equal slopes 2, different y-intercepts
∴ Lines (ii) 2y = 4x + 7 and (iv) 3y = 6x − 11 are parallel
Common mistakes that cost marks
- Reading the slope of 2y = 4x + 7 as 4. Divide by 2 first: the slope is 2.
- Giving the y-intercept of (iii) as −10. After dividing by 5 it is −2.
- Saying (ii) and (iv) are not parallel because the original equations look different. Compare slopes only after rewriting as y = ax + b.
How this can come in the exam
The slope and y-intercept of 4y = 2x − 12 are
- 2 and −12
- 12 and −3
- 2 and −3
- 12 and −12
Show answer
(B) 12 and −3
y = 12x − 3.
Assertion (A): The lines 3y = 9x + 2 and y = 3x − 5 are parallel.
Reason (R): Both lines cut the y-axis at the same point.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(C) A is true but R is false.
A is true: both slopes are 3. R is false: the y-intercepts are 23 and −5.
Try one yourself
Find the slope, the y-intercept and the point where 6y = −3x + 18 cuts the y-axis.
Show answer
y = −12x + 3: slope −12, y-intercept 3, point (0, 3).
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