Solving Systems by Elimination (Part 2)

ID: taguj-notov
Created by Illustrative MathIllustrative Math
Subject: Algebra, Algebra 2
Grade: 8-9

Solving Systems by Elimination (Part 2)

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Problem 1

1) Solve this system of linear equations without graphing: {5x + 4y = 810x  4y = 46\begin{cases} 5x \ + \ 4y \ = \ 8 \\ 10x \ - \ 4y \ = \ 46 \end{cases}

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Problem 2

2) Select all the equations that share a solution with this system of equations. Write each corresponding letter in the answer box and separate letters with commas.

{5x+4y=242x7y=26\begin{cases} 5x + 4y = 24 \\ 2x - 7y = 26 \end{cases}

a) 7x+3y=507x + 3y = 50 \quad \quad b) 7x3y=507x - 3y = 50 \quad \quad c) 5x+4y=2x7y5x + 4y = 2x - 7y \quad \quad d) 3x11y=23x - 11y = -2 \quad \quad e) 3x+11y=23x + 11y = -2

Problem 3

Students performed in a play on a Friday and a Saturday. For both performances, adult tickets cost aa dollars each and student tickets cost ss dollars each.

On Friday, they sold 125 adult tickets and 65 student tickets, and collected $1,200. On Saturday, they sold 140 adult tickets and 50 student tickets, and collect $1,230.

This situation is represented by this system of equations: {125a+65s=1,200140a+50s=1,230\begin{cases} 125a + 65s = 1,200 \\ 140a + 50s = 1,230 \end{cases}

3) What could the equation 265a+115s=2,430265a + 115s = 2,430 mean in this situation?

4) The solution to the original system is the pair a=7a = 7 and s=5s = 5. Explain why it makes sense that this pair of values is also the solution to the equation 265a+115s=2,430265a + 115s = 2,430.

Problem 4

5) Which statement explains why 13x13y=2613x - 13y = -26 shares a solution with this system of equations:

{10x3y=293x+10y=55\begin{cases} 10x - 3y = 29 \\ -3x + 10y = 55 \end{cases}

a) Because 13x13y=26 is the product of the two equations in the system of equations, it must share a solution with the system of equations.\text{Because } 13x - 13y = -26 \text{ is the product of the two equations in the system of equations, it must } \newline \text{share a solution with the system of equations.}b) The three equations all have the same slope but different y-intercepts. Equations with the same slope but different y-intercepts always share a solution.\text{The three equations all have the same slope but different } y \text{-intercepts. Equations with the same slope } \newline \text{but different } y \text{-intercepts always share a solution.}c) Because 10x3y is equal to 29, I can add 10x3y to the left side of 3x+10y=55 and add 29 to the right side of the same equation. Adding equivalent expressions to each side of an equation does not change the solution to the equation.\text{Because } 10x - 3y \text{ is equal to 29, I can add } 10x -3y \text{ to the left side of } -3x + 10y = 55 \text{ and add 29 } \newline \text{to the right side } \newline \text{of the same equation. Adding equivalent expressions to each side of an equation } \newline \text{does not change the solution to the equation.}d) Because 3x+10y is equal to 55, I can subtract 3x+10y from the left side of 10x3y=29 and subtract 55 from its right side. Subtracting equivalent expressions from each side of an equation does not change the solution to the equation.\text{Because } -3x + 10y \text{ is equal to 55, I can subtract } -3x + 10y \text{ from the left side of } 10x - 3y = 29 \text{ and } \newline \text{subtract 55 from its right side. Subtracting equivalent expressions from each side of an equation does } \newline \text{not change the solution to the equation.}
Problem 5

6) Select all equations that can result from adding these two equations or subtracting one from the other.

{x+y=123x5y=4\begin{cases} x + y = 12 \\ 3x - 5y = 4 \end{cases}

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

a) 2x4y=8-2x - 4y = 8 \quad \quad b) 2x+6y=8-2x + 6y = 8 \quad \quad c) 4x4y=164x - 4y = 16 \quad \quad d) 4x+4y=164x + 4y = 16 \quad \quad e) 2x6y=82x - 6y = -8 \quad \quad f) 5x4y=285x - 4y = 28

Problem 6

Solve each system of equations.

7) {7x12y=1807x=84\begin{cases} 7x - 12y = 180 \\ 7x = 84 \end{cases}

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8) {16y=4x4x+27y=11\begin{cases} -16y = 4x \\ 4x + 27y = 11 \end{cases}

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Problem 7

Here is a system of equations: {7x4y=117x+4y=59\begin{cases} 7x - 4y = -11 \\ 7x + 4y = -59 \end{cases}

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

10) Explain your reasoning.

Problem 8

The box plot represents the distribution of the number of free throws that 20 students made out of 10 attempts.

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

After reviewing the data, the value recorded as 1 is determined to have been an error. The box plot represents the distribution of the same data set, but with the minimum, 1, removed.

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

The median is 6 free throws for both plots.

11) Explain why the median remains the same when 1 was removed from the data set.

12) When 1 is removed from the data set, does the mean remain the same?

True or false? Write below.

13) Explain your reasoning.

Problem 9

In places where there are crickets, the outdoor temperature can be predicted by the rate at which crickets chirp. One equation that models the relationship between chirps and outdoor temperature is f=14c+40f = \frac{1}{4}c + 40, where cc is the number of chirps per minute and ff is the temperature in degrees Fahrenheit.

14) Suppose 110 chirps are heard in a minute. According to this model, what is the outdoor temperature?

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15) If it is 75^{\circ}F outside, about how many chirps can we expect to hear in one minute?

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16) The equation is only a good model of the relationship when the outdoor temperature is at least 55^{\circ}F. (Below that temperature, crickets aren't around or inclined to chirp.) How many chirps can we expect to hear in a minute at that temperature?

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17) On the coordinate plane, draw a graph that represents the relationship between the number of chirps and the temperature.

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18) Explain what the coefficient 14\frac{1}{4} in the equation tells us about the relationship.

19) Explain what the 40 in the equation tells us about the relationship.