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Additional Mathematics questions, chapter 3
Nama / NameKelas / ClassTarikh / Date

Systems of Equations

45 marks. Use Practice to check each answer as you go, or Exam to answer everything first. Show your working for Section B.

Section A: objective questions

12 marks
Choose the correct answer. Each question carries 1 mark.
1

Question 1

Which equation is a linear equation in three variables?
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C
A linear equation in three variables has the form ax + by + cz = d, with every variable to the power 1. x² and xy make an equation non-linear, and x + y = 10 has only two variables.
For the teacher: Remembering, low difficulty, standard 3.1.1
Textbook p. 71
2

Question 2

The three planes of a system of linear equations meet along a straight line. How many solutions does the system have?
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B
Every point on the line satisfies all three equations, so there are infinite solutions.
For the teacher: Remembering, low difficulty, standard 3.1.2
Textbook p. 73
3

Question 3

Solve x + y + z = 12, x − y = 2 and y − z = −1. What is the value of z?
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D
x = y + 2 and z = y + 1. So (y + 2) + y + (y + 1) = 12, 3y = 9, y = 3, and z = 4.
For the teacher: Applying, medium difficulty, standard 3.1.2
Textbook p. 74
4

Question 4

While eliminating variables from a system of three linear equations, a student obtains 0 = 5. What does this show?
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A
0 = 5 is never true, so no values satisfy all the equations. The planes do not meet. 0 = 0 would show infinite solutions.
For the teacher: Understanding, medium difficulty, standard 3.1.2
Textbook p. 75
5

Question 5

The sum of three numbers a, b and c is 24. The number a is twice b, and c is 4 more than b. What is the value of a?
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B
a = 2b and c = b + 4, so 2b + b + (b + 4) = 24, 4b = 20, b = 5. Then a = 10 and c = 9.
For the teacher: Applying, medium difficulty, standard 3.1.3
Textbook p. 77
6

Question 6

To solve x + y = 7 and x² + y² = 29 by substitution, which is the best first step?
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D
Make one variable the subject of the linear equation, then substitute into the non-linear one to get a quadratic equation.
For the teacher: Understanding, low difficulty, standard 3.2.1
Textbook p. 80
7

Question 7

Which pair is a solution of x − y = 1 and xy = 12?
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A
y = x − 1, so x² − x − 12 = 0, (x − 4)(x + 3) = 0. The solutions are (4, 3) and (−3, −4). Only (−3, −4) is listed.
For the teacher: Applying, medium difficulty, standard 3.2.1
Textbook p. 80
8

Question 8

Substitute y = x + 1 into x² + y² = 13. Which quadratic equation results, in its simplest form?
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C
x² + (x + 1)² = 13 gives 2x² + 2x + 1 = 13, so 2x² + 2x − 12 = 0. Dividing by 2: x² + x − 6 = 0.
For the teacher: Applying, low difficulty, standard 3.2.1
Textbook p. 80
9

Question 9

How many points of intersection do the graphs of y = x² and y = 2x − 5 have?
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A
x² = 2x − 5 gives x² − 2x + 5 = 0. The discriminant is (−2)² − 4(1)(5) = −16, which is negative, so the graphs do not meet.
For the teacher: Analysing, high difficulty, standard 3.2.1
Textbook p. 81
10

Question 10

A rectangular plank has perimeter 38 cm and area 90 cm². What is the length of its longer side?
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D
l + w = 19 and lw = 90. l(19 − l) = 90 gives l² − 19l + 90 = 0, (l − 9)(l − 10) = 0. The sides are 9 cm and 10 cm.
For the teacher: Applying, medium difficulty, standard 3.2.2
Textbook p. 84
11

Question 11

(3, 4) is a solution of y = x + 1 and x² + y² = 25. What is the other solution?
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C
x² + (x + 1)² = 25 gives x² + x − 12 = 0, so x = 3 or x = −4. When x = −4, y = −3.
For the teacher: Analysing, high difficulty, standard 3.2.1
Textbook p. 80
12

Question 12

Aiman says the system x + y + z = 3, 2x + 2y + 2z = 6 and x + y + z = 5 has infinite solutions because the first two equations are the same. Is he correct?
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B
x + y + z cannot equal both 3 and 5. The planes of equations 1 and 3 are parallel, so there is no point on all three planes.
For the teacher: Evaluating, high difficulty, standard 3.1.2
Textbook p. 75

Section B: structured questions

33 marks
Answer all questions. Show your working.
13

Question 13

Solve the system of linear equations using the elimination method.
[4 marks] (a)x + y + z = 3,   2x + y − z = 6,   x + 2y + 3z = 2
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x = 5, y = −3, z = 1
Tick each point you got.
For the teacher: Applying, medium difficulty, standard 3.1.2
Textbook p. 74
14

Question 14

Solve z = x + y, 2x − y + z = 9 and x + y + z = 16 using the substitution method.
[4 marks] (a)Find x, y and z.
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Tick each point you got.
For the teacher: Applying, medium difficulty, standard 3.1.2
Textbook p. 75
15

Question 15

A cultural village sold 36 tickets for RM1 310. The trekking package costs RM30, the optional package RM45 and the standard package RM55. The number of trekking tickets is twice the total of optional and standard tickets.
[2 marks] (a)Let t, o and s be the numbers of trekking, optional and standard tickets. Write three equations.
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Tick each point you got.
[3 marks] (b)Find the number of tickets sold for each package.
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24 trekking, 7 optional, 5 standard
Tick each point you got.
For the teacher: Applying, medium difficulty, standard 3.1.3
Textbook p. 68
16

Question 16

Determine whether each system has one solution, infinite solutions or no solution.
[2 marks] (a)x + y + z = 4,   2x + 2y + 2z = 8,   x − y = 0
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Tick each point you got.
[2 marks] (b)x + y + z = 4,   2x + 2y + 2z = 9,   x − y = 0
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Tick each point you got.
For the teacher: Analysing, high difficulty, standard 3.1.2
Textbook p. 75
17

Question 17

Diagram 1 shows the graphs of 2x + y = 4 and y² = 4x.
Graph of a straight line and a sideways parabola crossing at two points, one in the first quadrant and one in the fourth
Diagram 1
[2 marks] (a)Write the coordinates of the two points of intersection.
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Tick each point you got.
[3 marks] (b)Show by algebra that these are the solutions of the simultaneous equations.
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Tick each point you got.
For the teacher: Analysing, medium difficulty, standard 3.2.1
Textbook p. 82
18

Question 18

A rectangular garden has a fence of length 90 m around it and an area of 450 m².
[4 marks] (a)Find the length and width of the garden.
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Tick each point you got.
[1 mark] (b)Fencing costs RM12 per metre. Find the cost of the fence.
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Tick each point you got.
For the teacher: Applying, medium difficulty, standard 3.2.2
Textbook p. 84
19

Question 19

A cuboid has a square base of side x cm and height y cm. The total length of all its edges is 56 cm and its total surface area is 128 cm².
[2 marks] (a)Write two equations in x and y.
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Tick each point you got.
[2 marks] (b)Show that 3x² − 28x + 64 = 0.
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[2 marks] (c)Find the two possible volumes of the cuboid, correct to 1 decimal place where needed.
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Tick each point you got.
For the teacher: Analysing, high difficulty, standard 3.2.2
Textbook p. 84