Question

9.
the heights, in inches, of eight football players are given below.
76, 70, 72, 70, 69, 71, 78, 74
which box plot represents these data?
1)
2)
3)
4)
10.
the method of substitution was used to solve the system of equations below:
4x - 7y = 7
x - y = -1
which equation is a correct first step when using this method?

1. x = y - 1
2. y = x - 1
3. 3x - 6y = 8
4. 5x - 8y = 6

Explanation:

Problem 9 (Box Plot for Football Players' Heights)

Step 1: Order the Data

First, we order the data set: \( 69, 70, 70, 71, 72, 74, 76, 78 \).

Step 2: Find the Median (Q2)

Since there are 8 data points (even number), the median is the average of the 4th and 5th values.
\( \text{Median} = \frac{71 + 72}{2} = 71.5 \).

Step 3: Find Q1 (Median of Lower Half)

The lower half of the data is \( 69, 70, 70, 71 \). The median of this subset (Q1) is the average of the 2nd and 3rd values:
\( \text{Q1} = \frac{70 + 70}{2} = 70 \).

Step 4: Find Q3 (Median of Upper Half)

The upper half of the data is \( 72, 74, 76, 78 \). The median of this subset (Q3) is the average of the 2nd and 3rd values:
\( \text{Q3} = \frac{74 + 76}{2} = 75 \).

Step 5: Identify Min and Max

The minimum value is \( 69 \) and the maximum value is \( 78 \).

Now, we analyze the box plots:

• The box should span from Q1 (\( 70 \)) to Q3 (\( 75 \)), with the median at \( 71.5 \) inside the box.
• The whiskers should extend from min (\( 69 \)) to max (\( 78 \)).

Looking at the options, option 1 has the box from around 70 to 75, median around 71.5, whiskers from 69 to 78, which matches our calculations.

Step 1: Recall Substitution Method

The substitution method involves solving one equation for one variable and substituting it into the other equation.

Step 2: Analyze the Given Equations

We have the system:

$$
LATEXBLOCK0
$$

We can solve the second equation (\( x - y = -1 \)) for \( x \) (or \( y \)). Solving for \( x \):
\( x = y - 1 \) (by adding \( y \) to both sides and rearranging).

Let's check the options:

• Option 1: \( x = y - 1 \) (matches our derived equation).
• Option 2: \( y = x - 1 \) would be solving \( x - y = -1 \) for \( y \) as \( y = x + 1 \), which is incorrect.
• Option 3: \( 3x - 6y = 8 \) is not a valid first step (subtracting equations incorrectly).
• Option 4: \( 5x - 8y = 6 \) is also not a valid first step.

Answer:

1. The box plot labeled 1 (with number line 65, 70, 75, 80, 85, box from ~70 to ~75, median ~71.5, whiskers from ~69 to ~78)

Problem 10 (Substitution Method for System of Equations)