Mathematics questions, chapter 9

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Straight Lines

41 marks. Use Practice to check each answer as you go, or Exam to answer everything first. Do not read answers from the graphs unless a question says so: substitute and calculate. Show your working for Section B.

Section A: objective questions

12 marks

Choose the correct answer. Each question carries 1 mark.

Question 1

A straight line has the equation y = −3x + 7. What are its gradient and y-intercept?

gradient 7, y-intercept −3
gradient 3, y-intercept 7
gradient −3, y-intercept 7
gradient −3, y-intercept −7

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Compare y = −3x + 7 with y = mx + c: m = −3 and c = 7. The other options swap the two numbers or change a sign.

For the teacher: Remembering, low difficulty, standard 9.1.1

Textbook p. 227

Question 2

What is the gradient of the straight line 4y = 8x − 12?

2
8
−3
4

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Divide every term by 4 so that the coefficient of y is 1: y = 2x − 3. The gradient is 2. The number 8 is the gradient only if you forget to divide, and −3 is the y-intercept.

For the teacher: Understanding, low difficulty, standard 9.1.1

Textbook p. 229

Question 3

Diagram 1 shows a straight line on a Cartesian plane. What is the equation of the line?

A horizontal line parallel to the x-axis on a Cartesian plane with the x-axis and the y-axis marked from −4 to 4 — Diagram 1

x = −2
y = 2
y = −2x
y = −2

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The line is parallel to the x-axis and every point on it has y = −2, so its equation is y = −2. A vertical line would be x = −2, and y = −2x passes through the origin.

For the teacher: Understanding, low difficulty, standard 9.1.1

Textbook p. 229

Question 4

The straight line 2x + 3y = 12 is written in the form xa + yb = 1. What are the values of a and b?

a = 12, b = 12
a = 6, b = 4
a = 4, b = 6
a = 2, b = 3

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Divide every term by 12: 2x12 + 3y12 = 1, so x6 + y4 = 1. The x-intercept is a = 6 and the y-intercept is b = 4. The values 2 and 3 are only the coefficients, and 4 and 6 are in the wrong places.

For the teacher: Applying, low difficulty, standard 9.1.2

Textbook p. 231

Question 5

Write −x2 + y4 = 1 in the form y = mx + c.

y = −2x + 4
y = 2x + 4
y = 2x + 1
y = 12x + 4

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Multiply every term by 4: −2x + y = 4. Then y = 2x + 4. The option y = 2x + 1 forgets to multiply the 1 by 4, and y = −2x + 4 loses the sign when moving −2x to the right.

For the teacher: Applying, low difficulty, standard 9.1.2

Textbook p. 232

Question 6

Find the x-intercept and the y-intercept of the straight line 4x − 3y = 24.

x-intercept −8, y-intercept 6
x-intercept 6, y-intercept 8
x-intercept 4, y-intercept −3
x-intercept 6, y-intercept −8

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Put y = 0: 4x = 24, x = 6. Put x = 0: −3y = 24, y = −8. The option with y-intercept 8 forgets the negative sign.

For the teacher: Applying, low difficulty, standard 9.1.3

Textbook p. 235

Question 7

Diagram 2 shows the straight line x + 2y = 4 and four points K, L, M and N. Which point does not satisfy the equation of the line?

Line x + 2y = 4 passing through K(4, 0), L(2, 1) and N(−2, 3); point M(1, 2) is above the line — Diagram 2

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Read the coordinates and substitute into x + 2y. K(4, 0): 4. L(2, 1): 4. N(−2, 3): −2 + 6 = 4. M(1, 2): 1 + 4 = 5, which is not 4, and M is off the line.

For the teacher: Analysing, low difficulty, standard 9.1.3

Textbook p. 233

Question 8

The point (k, 5) lies on the straight line 2x − 3y = −7. What is the value of k?

4
−11
11
−4

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Substitute x = k and y = 5: 2k − 3(5) = −7, so 2k − 15 = −7 and 2k = 8, k = 4. The values −11 and 11 come from sign slips when moving −15.

For the teacher: Analysing, medium difficulty, standard 9.1.3

Textbook p. 235

Question 9

Which of the following straight lines is parallel to y = 3x − 5?

3x + y = 5
x − 3y = 5
y = −3x − 5
6x − 2y = 7

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Parallel lines have the same gradient, 3. 6x − 2y = 7 gives y = 3x − 72, gradient 3. 3x + y = 5 has gradient −3, x − 3y = 5 has gradient 13, and y = −3x − 5 has gradient −3.

For the teacher: Applying, low difficulty, standard 9.1.4

Textbook p. 237

Question 10

The straight lines 2x + 3y = 9 and 4x + ky = 5 are parallel. Find the value of k.

3
6
−6
12

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The first line has gradient −23. The second has gradient −4k. Equal gradients: 4k = 23, so k = 6.

For the teacher: Applying, medium difficulty, standard 9.1.4

Textbook p. 238

Question 11

A straight line has gradient −2 and passes through the point (3, 1). What is its equation?

y = −2x + 7
y = −2x − 5
y = −2x + 1
y = 2x − 5

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Substitute m = −2, x = 3, y = 1 into y = mx + c: 1 = (−2)(3) + c, so c = 7. The equation is y = −2x + 7.

For the teacher: Applying, low difficulty, standard 9.1.5

Textbook p. 239

Question 12

The straight lines y = 2x − 1 and x + y = 8 intersect at point P. What are the coordinates of P?

(5, 3)
(4, 7)
(3, 5)
(2, 6)

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Substitute y = 2x − 1 into x + y = 8: x + 2x − 1 = 8, so 3x = 9 and x = 3. Then y = 2(3) − 1 = 5. Check: 3 + 5 = 8. The other points satisfy only one of the equations.

For the teacher: Applying, low difficulty, standard 9.1.6

Textbook p. 242

Section B: structured questions

29 marks

Answer all questions. Show your working.

Question 13

State the gradient and y-intercept of each straight line.

[1 mark] (a)y = 5x − 2

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gradient = 5 and y-intercept = −2 [1]1

Tick each point you got.

[2 marks] (b)3y = −6x + 15

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y = −2x + 5 [1]1
gradient = −2 and y-intercept = 5 [1]1

Divide by 3 first so that the coefficient of y is +1.

Tick each point you got.

For the teacher: Understanding, low difficulty, standard 9.1.1

Textbook p. 229

Question 14

Diagram 3 shows a straight line that crosses the x-axis at one point and the y-axis at another.

A straight line rising to the right that crosses the x-axis and the y-axis at the two points marked with red dots — Diagram 3

[1 mark] (a)State the x-intercept and the y-intercept of the line.

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x-intercept = −3 and y-intercept = 6 [1]1

Tick each point you got.

[1 mark] (b)Calculate the gradient of the line.

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m = −6−3 = 2 [1]1

Gradient = −(y-intercept) ÷ (x-intercept).

Tick each point you got.

[1 mark] (c)Write the equation of the line in the form y = mx + c.

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y = 2x + 6 [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, standard 9.1.2

Textbook p. 231

Question 15

The equation of a straight line is 3x + 4y = 24.

[1 mark] (a)Write the equation in the form xa + yb = 1.

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x8 + y6 = 1 [1]1

Divide every term by 24 so that the right side is 1.

Tick each point you got.

[1 mark] (b)Write the equation in the form y = mx + c.

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y = −34x + 6 [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, standard 9.1.2

Textbook p. 231

Question 16

Answer each part. Show your substitution.

[1 mark] (a)Determine whether the point (3, −1) lies on the straight line 2x + 3y = 3.

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2(3) + 3(−1) = 3, equal to the right side, so the point lies on the line [1]1

Tick each point you got.

[1 mark] (b)Determine whether the point (−2, 6) lies on the straight line y = −2x + 1.

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−2(−2) + 1 = 5, which is not 6, so the point does not lie on the line [1]1

Tick each point you got.

[1 mark] (c)The point (4, h) lies on the straight line 5x − 2y = 8. Find the value of h.

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5(4) − 2h = 8, so 2h = 12 and h = 6 [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, standard 9.1.3

Textbook p. 234

Question 17

Hana says that the straight lines y = 3x + 2 and 6x − 2y = 5 are the same line because both have a gradient of 3.

[2 marks] (a)Find the gradient and the y-intercept of 6x − 2y = 5.

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y = 3x − 52 [1]1
gradient = 3 and y-intercept = −52 [1]1

Tick each point you got.

[1 mark] (b)Is Hana correct?

AHana is correctBHana is not correct

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Hana is not correct

The lines have the same gradient but different y-intercepts.

[1 mark] (c)Give a reason for your answer in (b).

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The y-intercepts are different (2 and −52), so the lines are parallel but not the same line [1]1

Tick each point you got.

For the teacher: Evaluating, medium difficulty, standard 9.1.4

Textbook p. 237

Question 18

Find the equation of each straight line.

[2 marks] (a)The line passes through the points (−2, −3) and (4, 9).

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m = 9 − (−3)4 − (−2) = 2 [1]1
9 = 2(4) + c, c = 1, so y = 2x + 1 [1]1

Tick each point you got.

[2 marks] (b)The line is parallel to 2x + y = 7 and passes through the point (3, −2).

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gradient = −2 [1]1
−2 = (−2)(3) + c, c = 4, so y = −2x + 4 [1]1

Tick each point you got.

For the teacher: Applying, medium difficulty, standard 9.1.5

Textbook p. 240

Question 19

Diagram 4 shows the straight lines x + y = 6 and y = 2x. They intersect at point A.

The lines x + y = 6 (falling) and y = 2x (rising) crossing at point A in the first quarter — Diagram 4

[1 mark] (a)State the coordinates of A from the graph.

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A = (2, 4) [1]1

Tick each point you got.

[2 marks] (b)Show, by solving the simultaneous equations x + y = 6 and y = 2x, that A is the point of intersection.

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x + 2x = 6, so 3x = 6 and x = 2 [1]1
y = 2(2) = 4, so the point is (2, 4) [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, standard 9.1.6

Textbook p. 242

Question 20

Diagram 5 shows a parallelogram PQRS on a Cartesian plane, where P(−4, 1), Q(2, 4) and R(6, 2). The straight line SR is parallel to PQ, and S lies on the y-axis.

Parallelogram PQRS with P at the left, Q at the top, R at the right and S on the y-axis below the x-axis — Diagram 5

[1 mark] (a)Calculate the gradient of PQ.

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m = 4 − 12 − (−4) = 12 [1]1

Tick each point you got.

[2 marks] (b)Find the equation of the straight line SR.

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gradient of SR = 12 [1]1
2 = 12(6) + c, c = −1, so y = 12x − 1 [1]1

SR is parallel to PQ, so it has the same gradient.

Tick each point you got.

[2 marks] (c)Find the equation of the straight line PS.

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S = (0, −1) (the y-intercept of SR) [1]1
gradient of PS = −1 − 10 − (−4) = −12, so y = −12x − 1 [1]1

Tick each point you got.

For the teacher: Analysing, high difficulty, standard 9.1.7

Textbook p. 245

Question 21

Create two different straight lines that intersect at the point (2, 5).

[2 marks] (a)Write the equations of two such straight lines and show that (2, 5) lies on both.

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one valid equation with (2, 5) shown to satisfy it, for example y = x + 3: 2 + 3 = 5 [1]1
a second, non-parallel equation with the check, for example x + y = 7: 2 + 5 = 7 [1]1

Any two lines through (2, 5) with different gradients are correct.

Tick each point you got.

For the teacher: Creating, medium difficulty, standard 9.1.6

Textbook p. 243

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