MPM2D Grade 10 a
Systems of Linear Equations Test # 2
Thinking, Inquiry & Problem Solving
Knowledge & Understanding
Application
Communication
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Knowledge
1. Solve the system of equations by substitution: [K 3]
aaa7x – y = 21,
(1)
aaa5y + 3x = – 29. (2)
Solution
From (1): y = 7x – 21 (3)
Substitute (3) into (2):
5(7x – 21) + 3x = – 29, 35x – 105 + 3x = – 29, 38x = 105 – 29,
38x = 76, x = 2.
Substitute x = 2 into (3): y = 7(2) – 21 = –7.
Answer: x = 2, y = –7.
Check:
1) 7x – y = 21
2) 5y + 3x = – 29
7(2) – (–7) = 21
5(–7) + 3(2) = – 29
14 + 7 = 21
– 35 + 6 = – 29
21= 21
– 29= – 29
2. Solve the system of equations by elimination: [K 3]
aaa8x – 3y = 22,
aaa6x + 12y = – 12.
Solution
8x – 3y = 22
× 2 16x – 6y = 44
6x + 12y = –12 ÷ 2 + 3x + 6y = –6
19x
= 38 ⟹ x = 2
Substitute into the first equation:
8(2) – 3y = 22, 16 – 3y = 22, –3y = 22 – 16, –3y = 6,
Answer: x = 2, y = –2.
y = –2.
Check:
1) 8x – 3y = 22
8(2) – 3(–2) = 22
16 + 6 = 22
22 = 22
2) 6x + 12y = –12
6(2) + 12(–2) = – 12
12 – 24 = – 12
– 12= – 12
3. Solve the system of equations: [K 4]
𝑥
𝑦–3
+
= 1,
9
3
0.5x – (y + 9) = 0.
Solution
Multiply (1) by 9:
𝑥
(1)
(2)
𝑦–3
9× +9×
=9×1
9
3
x + 3(y – 3) = 9
x + 3y – 9 = 9
x + 3y – 18 = 0
x = –3y + 18
(3)
Multiply (2) by 2:
2 × 0.5x – 2 × (y + 9) = 0
x – 2y – 18 = 0
(4)
Substitute (3) into (4):
––3y + 18 – 2y – 18 = 0
– 5y = 0
y=0
Substitute y = 0 into (3): x = –3(0) + 18 = 18.
Answer: x = 18, y = 0.
Check:
1)
𝑥
9
18
+
𝑦–3
=1
3
0–3
+
=1
3
2–1=1
1=1
9
2) 0.5x – (y + 9) = 0
0.5(18) – (0 + 9) = 0
9–9=0
0= 0
4. Without solving, determine how many solutions the following system of
equations has? Explain your answer. [K 3]
aaa2(x + 3) – 3(y + 5) = 0,
aaa8x = 38 + 12y.
Solution
2(x + 3) – 3(y + 5) = 0 ⟹ 2x + 6 – 3y – 15 = 0 ⟹ 2x – 3y = 9
8x = 38 + 12y ⟹ 8x – 12y = 38 ⟹ 4x – 6y = 19
2 −3
9
= ≠
∴ no solution
4 −6
19
2x – 3y = 9 and 4x – 6y = 19 are parallel lines, the do not intersect.
Answer: the system doesn’t have solutions.
Application
1. Jennifer is considering two job offers from shoe stores. Walk This Way offers a
monthly salary of $ 1 350 plus 2.5 % of any sales she makes. Petite Shoe offers a
monthly salary of $ 1 000 plus 3.5 % of any sales she makes.
a) When will both job offers produce the same salary? [A 4]
Solution
$x = monthly sales,
$S1 = monthly salary in Walk This Way, S1 = 0.025x + 1350
$S2 = monthly salary in Petite Shoe, S2 = 0.035x + 1000
S2 = S1 ⟹ 0.035x + 1000 = 0.025x + 1350
0.01x = 350 ⟹ x = $35 000.
Both job offers produce the same salary when monthly sales are $35 000.
b) What advice would you give Jenifer? [A 1]
Answer
If average monthly sales are less than $35 000, then Jennifer should accept Walk
This Way job offer, else she should accept Petite Shoe job offer.
2. Flying into the wind, an airliner takes 4h to go 960 km. The same plane flying
with the wind takes only 3h to make the same trip. Find the speed of the plane and
the speed of the wind. [A 4]
Solution
x km/h = speed of the airliner in still air
y km/h = speed of the wind
Speed (km/h)
Time (h) Distance (km)
with wind
x+y
3
3(x + y) = 960
against wind
x–y
4
4(x – y) = 960
3(x + y) = 960 ⟹ x + y = 320
4(x – y) = 960 ⟹ x – y = 240 +
2x = 560 ⟹ x = 280, y = 320 – x = 40 km/h
Answer: speed of the airliner in still air = 280 km/h,
speed of the wind = 40 km/h
3. Sandra is starting a lawn-cutting business. Her start-up cost to buy two lawn
mowers and an edge trimmer is $ 592. She has figured out that she will use about $
2 in gas for each lawn. If she charges $ 18 per lawn, what will her break-even point
be? [A 3]
Solution
n = number of lawns, P = profit in $.
P = (18 – 2) n – 592
= 16n – 592
P = 0 ⟹ 16n – 592 = 0, 16n = 592, n = 37.
Answer: Her break-even point is 37 lawns.
Thinking, Inquiry & Problem solving
1. The graphs of x – y – 3 = 0, 3x – y + 5 = 0, and kx – 2y – 2 = 0 all intersect at the
same point. Find the values of k. Show all the steps of tour solution. [T 3]
Solution
3x – y + 5 = 0 (2)
–
x – y – 3 = 0 (1)
2x
+ 8 = 0 ⟹ 2x = – 8, x = – 4.
Substitute x = – 4 into (1): – 4 – y – 3 = 0 ⟹ y = –7.
Substitute x = – 4 and y = –7 into the third equation:
kx – 2y – 2 = 0 ⟹ k(– 4) – 2(–7) – 2 = 0, 4k = 12, k = 3.
Answer: k = 3
2. The linear system 5x – 2y = – 6 and 8y – mx = n has an infinite number of
solutions. Determine the values of m and n. Justify your response. [T 3]
Solution Since the system has infinite number of solutions, the coefficients are
proportional:
5
−2
−6
=
=
8
5
−𝑚
−2
8
5
−𝑚
−6
1) =
𝑛
⟹
5
8
=
2
⟹m=
𝑚
8 ∙ (−6)
8∙2
5
= 3.2
2) = ⟹ n =
= – 9.6.
8
𝑛
5
Answer: m = 3.2, n = – 9.6.
Communication
1. At Sophie’s Java, a new blend of coffee is featured each week. This week, Sophie
is creating a low-caffeine espresso blend from Brazilian and Ethiopian beans. She
wants to make 100 kg of this blend and sell it for $13 / kg. The Brazilian beans sell
for $10 / kg and Ethiopian beans sell for $15 /kg. How many kilograms of each kind
of bean must Sophie use to make 100 kg of her new blend of the week? Model this
situation with linear system. Do not solve. [C 3]
Solution
Blends of coffee Mass (kg)
Brazilian
x
Ethiopian
y
Mixture
100
x + y = 100
Unit Price $/kg
10
15
13
Total cost
10x
15y
1300
10x + 15y = 1300
2. Explain, why the following linear system is not easy to solve by substitution:
[C 1]
3x + 4y = 10,
a.aa2x – 5y = 9.
Answer: None of the coefficients of the variables equals 1 or – 1. Therefore, to
solve this system by substitution, we must use fractions, which is not easy.
3. Determine, whether the ordered pair (– 4, 3) is the solution to the linear system
defined by 4y – 3x = 24 and 4x = – 10 – 2y. Justify your response. [C 3]
Answer:
Substitute x = – 4 and y = 3 into the equations:
4(3) – 3(– 4) = 24
4(– 4) = – 10 – 2(3)
12 + 12 = 24
–16 = – 10 – 6
24 = 24
–16 = – 16
Thus, the ordered pair (– 4, 3) is the solution to the linear system.
4. An air traffic controller is plotting the course of two jets scheduled to land in
about 10 minutes. One aircraft is following a path defined by the equation
4x – 2y = 14,
(1)
and the other by the equation
12x = 40 + 6y.
(2)
Should the controller alter the paths of each aircraft? Show all work and justify
your solution. [C 2]
Answer:
4x – 2y = 14 ⟹ – 2y = – 4x + 14 ⟹ y = 2x – 7
20
12x = 40 + 6y ⟹ 6y = 12x – 40 ⟹ y = 2x –
3
The planes follow parallel paths; therefore the controller should not alter the paths
of aircrafts.