Systems of Linear Equations: math practice, Grade 8

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Math practice Grade 8 : Systems of Linear Equations — Zyro the alien explorer of Planète Maths

22 problems graded by difficulty (★ easy, ★★ medium, ★★★ hard). Try first, without looking at the solutions!

Sommaire

1 Is it a solution? ★★★

Consider the system $ x + y = 5 $ and $ 2x - y = 4 $.

Is $ (3, 2) $ a solution?
Is $ (4, 1) $ a solution?

2 Table of values and graph ★★★

Solve the system $ y = x + 2 $ and $ y = -x + 4 $ by graphing. Make a table of values for $ x = 0, 1, 2, 3 $ for each line, graph them, and state the solution.

3 Substitute y = 2x ★★★

Solve by substitution: $ y = 2x $ and $ x + y = 12 $.

4 Add the equations ★★★

Solve by elimination: $ x + y = 11 $ and $ x - y = 3 $.

5 Substitute an expression ★★★

Solve by substitution: $ y = x - 3 $ and $ 2x + y = 15 $.

6 True or false? ★★★

Is this statement true or false? Justify. “A system of two different linear equations can have exactly two solutions.”

7 Predict the number of solutions ★★★

Without solving, decide whether each system has one solution, no solution, or infinitely many solutions.

$ y = 3x + 1 $ and $ y = 3x - 5 $
$ y = -x + 4 $ and $ y = 2x + 4 $
$ y = \dfrac{1}{2}x + 2 $ and $ 2y = x + 4 $

8 Elimination with one multiplier ★★★

Solve by elimination: $ 2x + 3y = 16 $ and $ 4x - y = 18 $.

9 Isolate first ★★★

Solve by substitution: $ x + 2y = 9 $ and $ 3x - y = 6 $. Hint: which variable is easiest to isolate?

10 Multiply both equations ★★★

Solve by elimination: $ 3x + 4y = -1 $ and $ 5x + 6y = -3 $.

11 A statement that is always true ★★★

Solve $ 4x - 2y = 6 $ and $ y = 2x - 3 $. What does the result tell you about the graphs?

12 A statement that is never true ★★★

Solve by elimination: $ 2x + y = 7 $ and $ 6x + 3y = 10 $. Explain the outcome.

13 Tickets at a school play ★★★

A school play sold 36 tickets. Adult tickets cost $8 and child tickets cost $5. The total was $243. How many tickets of each type were sold?

14 Find the missing slope ★★★

The system $ y = 3x + 4 $ and $ y = kx - 2 $ has no solution. Find $ k $. For what values of $ k $ does the system have exactly one solution?

15 Kayak in a river ★★★

A kayaker paddles 12 miles downstream in 2 hours and returns the same 12 miles upstream in 3 hours. Let $ b $ be the kayaker’s speed in still water and $ c $ the speed of the current, both in miles per hour. Find $ b $ and $ c $.

16 Two numbers ★★★

The sum of two numbers is 54. One number is 6 more than twice the other. Find the numbers.

17 Trail mix ★★★

Cashews cost $9 per pound and raisins cost $4 per pound. How many pounds of each should you mix to make 10 pounds of trail mix that costs $6 per pound? (1 pound is about 0.45 kilogram.)

18 Intersection of two lines through points ★★★

Line $ p $ goes through $ (0, 1) $ and $ (2, 5) $. Line $ q $ goes through $ (0, 9) $ and $ (3, 0) $. Find the equation of each line and the point where they meet.

19 Fractions in a system ★★★

Solve $ \dfrac{x}{2} + \dfrac{y}{3} = 4 $ and $ x - y = 3 $. Hint: clear the denominators first.

20 Choosing a plan ★★★

A climbing gym offers two plans. Plan A costs $15 to join plus $2 per visit, so $ y = 2x + 15 $. Plan B costs $5 to join plus $3 per visit, so $ y = 3x + 5 $. Here $ x $ is the number of visits and $ y $ the total cost in dollars.

Find the point where the plans cost the same and explain what it means.
Which plan is cheaper for 6 visits? For 15 visits?

21 A race with a head start ★★★

Sam starts 200 meters ahead of Mia and runs at 3 meters per second. Mia starts at the starting line and runs at 5 meters per second. After $ t $ seconds, Sam is at $ d = 200 + 3t $ meters and Mia at $ d = 5t $ meters. When does Mia catch Sam, and how far has she run? (1 meter is about 3.28 feet.)

22 Find the error ★★★

Lena solves $ 5x - 2y = 11 $ and $ 3x + 2y = 5 $. She subtracts the second equation from the first and writes $ 2x = 6 $, so $ x = 3 $. Find her mistake, then solve the system correctly.