Ontario Grade 10 Math

Unit 1 · Ontario MPM2D

Ontario Grade 10 Linear Systems Practice Questions

Two equations, two unknowns — find where they meet. Practise substitution, elimination and graphing. Word problems add one step: name the variables, write the two equations, then solve. Try on paper first, then check the steps.

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Practice questions with solutions

13 questions. Try each one on paper, then open the worked solution.

Question 1 · Linear Systems Concepts

Is the following statement true or false? "If two equations in a linear system represent the same line, the system has infinitely many solutions."
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If the equations represent the same line (coincident lines), they intersect at every point along the line, yielding infinitely many solutions. The answer is a.

Answer: True

Question 2 · Solving by Graphing

Solve the linear system by graphing.y=2x−1y = 2x - 1y=−x+5y = -x + 5What is the point of intersection?
x y line 1 line 2
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Graph y = 2x - 1: y-intercept at (0,-1), slope 2. Graph y = -x + 5: y-intercept at (0,5), slope -1. The lines intersect at (2, 3). Check: 3 = 2(2) - 1 = 3 ✓ 3 = -(2) + 5 = 3 ✓

Answer: (2,3)

Question 3 · Solving by Graphing

Solve the linear system by graphing.y=12x+2y = \frac{1}{2}x + 2y=2x−1y = 2x - 1What is the point of intersection?
x y line 1 line 2
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Graph y = ½x + 2: y-intercept (0,2), slope ½ (rise 1, run 2). Graph y = 2x - 1: y-intercept (0,-1), slope 2 (rise 2, run 1). The lines intersect at (2, 3). Check: 3 = ½(2) + 2 = 3 ✓ 3 = 2(2) - 1 = 3 ✓

Answer: (2,3)

Question 4 · Solving by Graphing

Solve the linear system by graphing.x+y=4x + y = 42x−y=−12x - y = -1What is the point of intersection?
x y line 1 line 2
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Rewrite each in slope-intercept form. x + y = 4 → y = -x + 4: y-intercept (0,4), slope -1. 2x - y = -1 → y = 2x + 1: y-intercept (0,1), slope 2. The lines intersect at (1, 3). Check: 1 + 3 = 4 ✓ 2(1) - 3 = -1 ✓

Answer: (1,3)

Question 5 · Solving by Elimination

Solve the following linear system using elimination:5x+2y=−115x + 2y = -113x+2y=−93x + 2y = -9
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Subtract equation 2 from equation 1: (5x + 2y) - (3x + 2y) = -11 - (-9) 2x = -2 x = -1 Then 3(-1) + 2y = -9 → 2y = -6 → y = -3 Solution: (-1, -3)

Answer: (-1,-3)

Question 6 · Solving by Elimination

Solve the following linear system using elimination:7x+3y=−177x + 3y = -176x+2y=−146x + 2y = -14
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Multiply equation 1 by 2: 14x + 6y = -34 Multiply equation 2 by 3: 18x + 6y = -42 Subtract: 4x = -8 → x = -2 Then 7(-2) + 3y = -17 → 3y = -3 → y = -1 Solution: (-2, -1)

Answer: (-2,-1)

Question 7 · Solving by Elimination

Solve the following linear system using elimination:5x+7y=35x + 7y = 32x+3y=12x + 3y = 1
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Multiply equation 1 by 2: 10x + 14y = 6 Multiply equation 2 by 5: 10x + 15y = 5 Subtract: -y = 1 → y = -1 Then 2x + 3(-1) = 1 → 2x = 4 → x = 2 Solution: (2, -1)

Answer: (2,-1)

Question 8 · Solving by Substitution

Solve the following linear system using substitution:y=2x−3y = 2x - 3x+y=6x + y = 6
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Substitute y = 2x - 3 into x + y = 6: x + (2x - 3) = 6 3x - 3 = 6 3x = 9 x = 3 Then y = 2(3) - 3 = 3 Solution: (3, 3)

Answer: (3,3)

Question 9 · Solving by Substitution

Solve the following linear system using substitution:2x+y=62x + y = 63x+2y=103x + 2y = 10
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From equation 1: y = 6 - 2x Substitute into equation 2: 3x + 2(6 - 2x) = 10 3x + 12 - 4x = 10 -x = -2 x = 2 Then y = 6 - 2(2) = 2 Solution: (2, 2)

Answer: (2,2)

Question 10 · Solving by Substitution

Solve the following linear system using substitution:4x+y=04x + y = 0x+2y+1=0x + 2y + 1 = 0
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From equation 1: y = -4x Substitute into equation 2: x + 2(-4x) + 1 = 0 x - 8x + 1 = 0 -7x = -1 x = 1/7 Then y = -4(1/7) = -4/7 Solution: (1/7, -4/7)

Answer: (1/7,-4/7)

Question 11 · Linear Systems Word Problems

Three soccer balls and a basketball cost $155. Two soccer balls and three basketballs cost $220. Find the cost of each ball.
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Let s be the cost of a soccer ball and b be the cost of a basketball. Formula/model: 3s + b = 155 2s + 3b = 220 Solve: From the first equation, b = 155 - 3s. Substitute into the second equation: 2s + 3(155 - 3s) = 220 → -7s = -245 → s = 35. Then b = 155 - 3(35) = 50. Therefore, soccer=$35, basketball=$50

Answer: soccer=$35, basketball=$50

Question 12 · Linear Systems Word Problems

The Sports Shop sells Adidas running shoes for $82 a pair and Air Jensen basketball shoes for $95 a pair. One day, the shop sells a combined 75 pairs totaling $6241 in sales. How many pairs of each shoe were sold?
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Let a be the number of Adidas pairs and j be the number of Jensen pairs. Formula/model: a + j = 75 82a + 95j = 6241 Solve: From the first equation, a = 75 - j. Substitute into the second equation: 82(75 - j) + 95j = 6241 → 13j = 91 → j = 7. Then a = 75 - 7 = 68. Therefore, Adidas=68, Jensen=7

Answer: Adidas=68, Jensen=7

Question 13 · Linear Systems Word Problems

A blue spruce tree grows an average of 15 cm per year. An eastern hemlock grows an average of 10 cm per year. When planted, the blue spruce was 120 cm tall and the eastern hemlock was 180 cm tall. How many years after planting will the trees reach the same height?
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Let y be the number of years after planting. Formula/model: 120 + 15y = 180 + 10y Solve: 5y = 60 → y = 12. The height is 120 + 15(12) = 300. Therefore, 12 years, 300 cm

Answer: 12 years, 300 cm

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