# Lesson 15

Solving Systems by Elimination (Part 2)

• Let’s think about why adding and subtracting equations works for solving systems of linear equations.

### Problem 1

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

### Problem 2

Select all the equations that share a solution with this system of equations.

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

A:

$$7x + 3y = 50$$

B:

$$7x - 3y = 50$$

C:

$$5x + 4y = 2x - 7y$$

D:

$$3x - 11y = \text -2$$

E:

$$3x + 11y = \text -2$$

### Problem 3

Students performed in a play on a Friday and a Saturday. For both performances, adult tickets cost $$a$$ dollars each and student tickets cost $$s$$ 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: $$\begin{cases} 125a + 65s = 1,\!200 \\ 140a + 50s = 1,\!230 \\ \end{cases}$$

1. What could the equation $$265a + 115s = 2,\!430$$ mean in this situation?
2. The solution to the original system is the pair $$a=7$$ and $$s=5$$. Explain why it makes sense that this pair of values is also the solution to the equation $$265a + 115s = 2,\!430$$.

### Problem 4

Which statement explains why $$13x-13y = \text-26$$ shares a solution with this system of equations: $$\begin{cases} 10x - 3y = 29 \\ \text -3x + 10y = 55 \\ \end{cases}$$

A:

Because $$13x - 13y = \text -26$$ is the product of the two equations in the system of equations, it the must 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.

C:

Because $$10x - 3y$$ is equal to 29, I can add $$10x - 3y$$ to the left side of $$\text -3x + 10y = 55$$ 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.

D:

Because $$\text -3x + 10y$$ is equal to 55, I can subtract $$\text -3x + 10y$$ from the left side of $$10x - 3y = 29$$ and subtract 55 from its right side. Subtracting equivalent expressions from each side of an equation does not change the solution to the equation.

### Problem 5

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

$$\displaystyle \begin{cases} x+y=12 \\ 3x-5y=4 \\ \end{cases}$$

A:

$$\text-2x-4y=8$$

B:

$$\text-2x+6y=8$$

C:

$$4x-4y=16$$

D:

$$4x+4y=16$$

E:

$$2x-6y=\text-8$$

F:

$$5x-4y=28$$

(From Unit 2, Lesson 14.)

### Problem 6

Solve each system of equations.

1. $$\begin{cases} 7x-12y=180 \\ 7x=84 \\ \end{cases}$$

2. $$\begin{cases}\text-16y=4x\\ 4x+27y=11\\ \end{cases}$$

(From Unit 2, Lesson 13.)

### Problem 7

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

Would you rather use subtraction or addition to solve the system? Explain your reasoning.

(From Unit 2, Lesson 14.)

### Problem 8

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

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.

The median is 6 free throws for both plots.

1. Explain why the median remains the same when 1 was removed from the data set.
2. When 1 is removed from the data set, does the mean remain the same? Explain your reasoning.
(From Unit 1, Lesson 10.)

### 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 $$f = \frac14 c + 40$$, where $$c$$ is the number of chirps per minute and $$f$$ is the temperature in degrees Fahrenheit.

1. Suppose 110 chirps are heard in a minute. According to this model, what is the outdoor temperature?
2. If it is $$75^{\circ}F$$ outside, about how many chirps can we expect to hear in one minute?
3. 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?
5. Explain what the coefficient $$\frac14$$ in the equation tells us about the relationship.