Solving Systems by Elimination (Part 1)

ID: donub-duram
Created by Illustrative MathIllustrative Math
Subject: Algebra, Algebra 2
Grade: 8-9

Solving Systems by Elimination (Part 1)

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Problem 1

1) Which equation is the result of adding these two equations?

{2x+4y=173x10y=3\begin{cases} -2x + 4y = 17 \\ 3x - 10y = -3 \end{cases}

a) 5x6y=14 -5x - 6y = 14 b) x6y=14 -x - 6y = 14 c) x6y=14 x - 6y = 14 d) 5x+14y=20 5x + 14y = 20
Problem 2

2) Which equation is the result of subtracting the second equation from the first?

{4x6y=135x+2y=5\begin{cases} 4x - 6y = 13 \\ -5x + 2y = 5 \end{cases}

a) 9x4y=8 -9x - 4y = 8 b) x+4y=8 -x + 4y = 8 c) x4y=8 x - 4y = 8 d) 9x8y=8 9x - 8y = 8
Problem 3

3) Solve this system of equations without graphing: {5x+2y=295x2y=41\begin{cases} 5x + 2y = 29 \\ 5x - 2y = 41 \end{cases}

Show Work
Problem 4

Here is a system of linear equations: {6x+21y=1036x+23y=51\begin{cases} 6x + 21y = 103 \\ -6x + 23y = 51 \end{cases}

4) Would you rather use subtraction or addition to solve the system?

5) Explain your reasoning.

Problem 5

6) Kiran sells ff full boxes and hh half-boxes of fruit to raise money for a band trip. He earns $5 for each full box and $2 for each half-box of fruit he sells and earns a total of $100 toward the cost of his band trip. The equation 5f+2h=1005f + 2h = 100 describes this relationship.

Solve the equation for ff.

Show Work
Problem 6

Match each equation with the corresponding equation solved for aa. Write the letter of the correct equation in each answer box.

a) a=2b5a = \frac{2b}{5} \quad \quad b) a=2b5a = -\frac{2b}{5} \quad \quad c) a=2ba = -2b \quad \quad d) a=2b5a = 2b - 5 \quad \quad e) a=52ba = 5 - 2b

7) a+2b=5a + 2b = 5

8) 5a=2b5a = 2b

9) a+5=2ba + 5 = 2b

10) 5(a+2b)=05(a + 2b) = 0

11) 5a+2b=05a + 2b = 0

Problem 7

The volume of a cylinder is represented by the formula V=πr2hV = \pi r^2 h.

Find the value of each labelled cell in the table.

volume (cubic inches)radius (inches)height (inches)96π4A31.25π2.5BVrC\begin{array}{|c|c|c|} \hline \\[-1em] \textbf{volume (cubic inches)} & \textbf{radius (inches)} & \textbf{height (inches)} \\[-1em] \\ \hline \\[-1em] 96 \pi & 4 & \text{A} \\[-1em] \\ \hline \\[-1em] 31.25 \pi & 2.5 & \text{B} \\[-1em] \\ \hline \\[-1em] V & r & \text{C} \\[-1em] \\ \hline \end{array}

12) Cell A

13) Cell B

14) Cell C

Problem 8

Match each equation with the slope mm and yy-intercept of its graph. Write the letter of the correct equation in each answer box.

a) 5x6y=305x - 6y = 30 \quad \quad b) y=56xy = 5 - 6x \quad \quad c) y=56x+1y = \frac{5}{6}x + 1 \quad \quad d) 5x6y=65x - 6y = 6 \quad \quad e) 5x+6y=65x + 6y = 6 \quad \quad f) 6x+y=126x + y = 12

15) m=6, yint=(0,12)m = -6, \ y-\text{int} = (0,12)

16) m=6, yint=(0,5)m = -6, \ y-\text{int} = (0,5)

17) m=56, yint=(0,1)m = -\frac{5}{6}, \ y-\text{int} = (0,1)

18) m=56, yint=(0,1)m = \frac{5}{6}, \ y-\text{int} = (0,1)

19) m=56, yint=(0,1)m = \frac{5}{6}, \ y-\text{int} = (0,-1)

20) m=56, yint=(0,5)m = \frac{5}{6}, \ y-\text{int} = (0,-5)

Problem 9

Solve each system of equations.

21) {2x+3y=42x=7y+24\begin{cases} 2x + 3y = 4 \\ 2x = 7y + 24 \end{cases}

Show Work

22) {5x+3y=233y=15x21\begin{cases} 5x + 3y = 23 \\ 3y = 15x - 21 \end{cases}

Show Work
Problem 10

Elena and Kiran are playing a board game. After one round, Elena says, "You earned so many more points than I did. If you earned 5 more points, your score would be twice mine!"

Kiran says, "Oh, I don't think I did that much better. I only scored 9 points higher than you did."

23) Write a system of equations to represent each student's comment. Use the variables kk and ee.

24) Specify what each variable represents.

25) If both students were correct, how many points did Elena score?

26) Show your reasoning.

27) If both students were correct, how many points did Kiran score?

28) Show your reasoning.