Solving Systems by Elimination (Part 3)

ID: gariv-firam
Created by Illustrative MathIllustrative Math
Subject: Algebra, Algebra 2
Grade: 8-9

Solving Systems by Elimination (Part 3)

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Problem 1

Solve each system of equations.

1) {2x4y=10x+5y=40\begin{cases} 2x - 4y = 10 \\ x + 5y = 40 \end{cases}

Show Work

2) {3x5y=42x+6y=18\begin{cases} 3x - 5y = 4 \\ -2x + 6y = 18 \end{cases}

Show Work
Problem 2

Tyler is solving this system of equations: {4p+2q=628pq=59\begin{cases} 4p + 2q = 62 \\ 8p - q = 59 \end{cases}

He can think of two ways to eliminate a variable and solve the system:

  • Multiply 4p+2q=624p + 2q = 62 by 2, then subtract 8pq=598p - q = 59 from the result.
  • Multiply 8pq=598p - q = 59 by 2, then add the result to 4p+2q=624p + 2q = 62.

3) Do both strategies work for solving the system?

True or false? Write below.

4) Explain or show your reasoning.

Problem 3

Andre and Elena are solving this system of equations: {y=3xy=9x30\begin{cases} y = 3x \\ y = 9x - 30 \end{cases}

  • Andre's first step is to write: 3x=9x303x = 9x - 30
  • Elena’s first step is to create a new system: {3y=9xy=9x30\begin{cases} 3y = 9x \\ y = 9x - 30 \end{cases}

5) Do you agree with either first step?

6) Explain your reasoning.

Problem 4

7) Select all systems that are equivalent to this system: {6d+4.5e=16.55d+0.5e=4\begin{cases} 6d + 4.5e = 16.5 \\ 5d + 0.5e = 4 \end{cases}

Write each corresponding letter in the answer box and separate letters with commas.

a) {6d+4.5e=16.545d+4.5e=4\begin{cases} 6d + 4.5e = 16.5 \\ 45d + 4.5e = 4 \end{cases} \quad \quad b) {30d+22.5e=82.55d+0.5e=4\begin{cases} 30d + 22.5e = 82.5 \\ 5d + 0.5e = 4 \end{cases} \quad \quad c) {30d+22.5e=82.530d+3e=24\begin{cases} 30d + 22.5e = 82.5 \\ 30d + 3e = 24 \end{cases}

d) {6d+4.5e=16.56d+0.6e=4.8\begin{cases} 6d + 4.5e = 16.5 \\ 6d + 0.6e = 4.8 \end{cases} \quad \quad e) {12d+9e=3310d+0.5e=8\begin{cases} 12d + 9e = 33 \\ 10d + 0.5e = 8 \end{cases} \quad \quad \quad f) {6d+4.5e=16.511d+5e=20.5\begin{cases} 6d + 4.5e = 16.5 \\ 11d + 5e = 20.5 \end{cases}

Problem 5

Here is a system of equations with a solution: {p+8q=812p+5q=5\begin{cases} p + 8q = -8 \\ \frac{1}{2}p + 5q = -5 \end{cases}

8) Write a system of equations that is equivalent to this system.

9) Describe what you did to the original system to get the new system.

10) Explain how you know the new system has the same solution as the original system.

Problem 6

The cost to mail a package is $5.00. Noah has postcard stamps that are worth $0.34 each and first-class stamps that are worth $0.49 each.

11) Write an equation that relates the number of postcard stamps pp, the number of first-class stamps ff, and the cost of mailing the package.

12) Solve the equation for ff.

Show Work

13) Solve the equation for pp.

Show Work

14) If Noah puts 7 first-class stamps on the package, how many postcard stamps will he need?

Show Work
Problem 7

Here is a system of linear equations: {2x+7y=8y+2x=14\begin{cases} 2x + 7y = 8 \\ y + 2x = 14 \end{cases}

15) Find at least one way to solve the system by substitution.

16) Show your reasoning.

17) How many ways can you find? (Regardless of the substitution that you do, the solution should be the same.)

Problem 8

18) Here is a system of equations: {7x+3y=657x+10y=135\begin{cases} -7x + 3y = -65 \\ -7x + 10y = -135 \end{cases}

Write an equation that results from subtracting the two equations.

Problem 9

19) A grocery store sells bananas for bb dollars a pound and grapes for gg dollars a pound. Priya buys 2.2 pounds of bananas and 3.6 pounds of grapes for $9.35. Andre buys 1.6 pounds of bananas and 1.2 pounds of grapes for $3.68.

This situation is represented by the system of equations: {2.2b+3.6g=9.351.6b+1.2g=3.68\begin{cases} 2.2b + 3.6g = 9.35 \\ 1.6b + 1.2g = 3.68 \end{cases}

Explain why it makes sense in this situation that the solution of this system is also a solution to 3.8b+4.8g=13.033.8b + 4.8g = 13.03.