Solving Systems by Substitution

ID: tudap-dusif
Created by Illustrative MathIllustrative Math
Subject: Algebra, Algebra 2
Grade: 8-9

Solving Systems by Substitution

Classroom:
Due:
Student Name:
Date Submitted:
Problem 1

1) Identify a solution to this system of equations: {4x+3y=23xy=7\begin{cases} -4x + 3y = 23 \\ x - y = -7 \end{cases}

a) (-5, 2)\text{(-5, 2)}b) (-2, 5)\text{(-2, 5)}c) (-3, 4)\text{(-3, 4)}d) (4, -3)\text{(4, -3)}
Problem 2

Lin is solving this system of equations: {6x5y=343x+2y=8\begin{cases} 6x - 5y = 34 \\ 3x + 2y = 8 \end{cases}

She starts by rearranging the second equation to isolate the yy variable: y=41.5xy = 4 - 1.5x. She then substituted the expression 41.5x4 - 1.5x for yy in the first equation, as shown:

6x5(41.5x)=346x207.5x=341.5x=54x=36y=41.5xy=41.5(36)y=58\begin{array}{cc}\begin{array}{rcl} \\[-1em] 6x - 5(4 - 1.5x) & = & 34 \\[-1em] \\ \\[-1em] 6x - 20 - 7.5x & = & 34 \\[-1em] \\ \\[-1em] -1.5x & = & 54 \\[-1em] \\ \\[-1em] x & = & -36 \\[-1em] \end{array} & \begin{array}{rcl} \\[-1em] y & = & 4 - 1.5x \\[-1em] \\ \\[-1em] y & = & 4 - 1.5 \cdot (-36) \\[-1em] \\ \\[-1em] y & = & 58 \\[-1em] \end{array} \end{array}

2) Check to see if Lin's solution of (-36, 58) makes both equations in the system true.

True or false? Write below.
Show Work

3) If your answer to the previous question is "no," find and explain her mistake. If your answer is "yes," graph the equations to verify the solution of the system.

Problem 3

Solve each system of equations.

4) {2x4y=20x=4\begin{cases} 2x - 4y = 20 \\ x = 4 \end{cases}

Show Work

5) {y=6x+112x3y=7\begin{cases} y = 6x + 11 \\ 2x - 3y = 7 \end{cases}

Show Work
Problem 4

6) Tyler and Han are trying to solve this system by substitution: {x+3y=59x+3y=3\begin{cases} x + 3y = -5 \\ 9x + 3y = 3 \end{cases}

Tyler's first step is to isolate xx in the first equation to get x=53yx = -5 - 3y. Han's first step is to isolate 3y3y in the first equation to get 3y=5x3y = -5 - x.

Show that both first steps can be used to solve the system and will yield the same solution.

Problem 5

The dot plots show the distribution of the length, in centimeters, of 25 shark teeth for an extinct species of shark and the length, in centimeters, of 25 shark teeth for a closely related shark species that is still living.

A template for answering this question. Ask your instructor for an alternative.

7) Compare the two dot plots using the shape of the distribution. Use the situation described in the problem in your explanation.

8) Compare the two dot plots using the measures of center. Use the situation described in the problem in your explanation.

9) Compare the two dot plots using the measures of variability. Use the situation described in the problem in your explanation.

Problem 6

10) Kiran buys supplies for the school’s greenhouse. He buys ff bags of fertilizer and pp packages of soil. He pays $5 for each bag of fertilizer and $2 for each package of soil, and spends a total of $90. The equation 5f+2p=905f + 2p = 90 describes this relationship.

If Kiran solves the equation for pp, which equation would result?

a) 2p=905f 2p = 90 - 5f b) p=5f902 p = \frac{5f - 90}{2} c) p=452.5f p = 45 - 2.5f d) p=85f2 p = \frac{85f}{2}
Show Work
Problem 7

Elena wanted to find the slope and yy-intercept of the graph of 25x20y=10025x - 20y = 100. She decided to put the equation in slope-intercept form first. Here is her work:

25x20y=10020y=10025xy=554x\begin{array}{rcl} \\[-1em] 25x - 20y & = & 100 \\[-1em] \\ \\[-1em] 20y & = & 100 - 25x \\[-1em] \\ \\[-1em] y & = & 5 - \frac{5}{4}x \\[-1em] \end{array}

She concluded that the slope is 54-\frac{5}{4} and the yy-intercept is (0,5)(0, 5).

11) What was Elena’s mistake?

12) What is the slope of the line?

13) Explain or show your reasoning.

14) What are the yy-intercept of the line?

15) Explain or show your reasoning.

Problem 8

Find the xx- and yy-intercepts of the graph of each equation.

y=102xy = 10 - 2x

16) Find the xx-intercept of the graph.

17) Find the yy-intercept of the graph.

4y+9x=184y + 9x = 18

18) Find the xx-intercept of the graph.

19) Find the yy-intercept of the graph.

6x2y=446x - 2y = 44

20) Find the xx-intercept of the graph.

21) Find the yy-intercept of the graph.

2x=4+12y2x = 4 + 12y

22) Find the xx-intercept of the graph.

23) Find the yy-intercept of the graph.

Problem 9

24) Andre is buying snacks for the track and field team. He buys aa pounds of apricots for $6 per pound and bb pounds of dried bananas for $4 per pound. He buys a total of 5 pounds of apricots and dried bananas and spends a total of $24.50.

Which system of equations represents the constraints in this situation?

a) {6a+4b=5a+b=24.50 \begin{cases} 6a + 4b = 5 \\ a + b = 24.50 \end{cases} b) {6a+4b=24.50a+b=5 \begin{cases} 6a + 4b = 24.50 \\ a + b = 5 \end{cases} c) {6a=4b5(a+b)=24.50 \begin{cases} 6a = 4b \\ 5(a + b) = 24.50 \end{cases} d) {6a+b=45a+b=24.50 \begin{cases} 6a + b = 4 \\ 5a + b = 24.50 \end{cases}
Problem 10

Here are two equations:

Equation 1: y=3x+8y = 3x + 8

Equation 2: 2xy=62x - y = -6

Without using graphing technology:

25) Find a point that is a solution to Equation 1 but not a solution to Equation 2.

Show Work

26) Find a point that is a solution to Equation 2 but not a solution to Equation 1.

Show Work

27) Graph the two equations.

A template for answering this question. Ask your instructor for an alternative.

28) Find a point that is a solution to both equations.