Question

Use the table to answer the question.

x
y

6 16
15 43
Write the equation of the linear function that models the relationship shown in the table. Enter your answer in y=mx+b
form.

Answer 1

Last year, Alayah ordered 4 cubic yards of soil and spent $220. This year, she orders 7 cubic yards of soil and spends $340. Write the equation of the linear function that models the relationship between the number of cubic yards of soil ordered and the total cost. Enter your answer in y=mx+b
form.(1 point)

Answer 2

Compare Linear Function 1 with Linear Function 2. Which one has the greater rate of change? Choose 1 for Linear Function 1; choose 2 for Linear Function 2.

Linear Function 1: y=x+8

Linear Function 2: A coordinate plane shows the x-axis ranging from negative 2 to 6 in increments of 1 and the y-axis ranging from negative 2 to 12 in increments of 1. A line with arrows at both ends joins two plotted points. The coordinates of the plotted points are as follows: left parenthesis 2 comma 6 right parenthesis and left parenthesis 4 comma 10 right parenthesis.

Answer 3

To find the equation of the linear function in the form \(y = mx + b\), we can use the two given points from the table: \((6, 16)\) and \((15, 43)\).

1. **Calculate the slope (m)** using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
where \((x_1, y_1) = (6, 16)\) and \((x_2, y_2) = (15, 43)\).

Plugging in the values:
\[
m = \frac{43 - 16}{15 - 6} = \frac{27}{9} = 3
\]

2. **Use the slope (m) and one of the points to find the y-intercept (b)**. We can use the point \((6, 16)\):
\[
y = mx + b
\]
Substituting \(m = 3\), \(x = 6\), and \(y = 16\):
\[
16 = 3(6) + b
\]
\[
16 = 18 + b
\]
\[
b = 16 - 18 = -2
\]

3. **Write the equation of the linear function**:
\[
y = 3x - 2
\]

Thus, the equation of the linear function that models the relationship shown in the table is:
\[
y = 3x - 2
\]

Answer 4

To find the equation of the linear function that models the relationship between the number of cubic yards of soil ordered (let's denote this as \( x \)) and the total cost (denote as \( y \)), we can use the two data points provided:

1. Last year: \( (4, 220) \)
2. This year: \( (7, 340) \)

### Step 1: Calculate the Slope (m)

Using the formula for the slope \( m \):
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the points \( (x_1, y_1) = (4, 220) \) and \( (x_2, y_2) = (7, 340) \):
\[
m = \frac{340 - 220}{7 - 4} = \frac{120}{3} = 40
\]

### Step 2: Use the Slope (m) and a Point to Find the y-Intercept (b)

We can use the slope \( m = 40 \) and one of the points to find \( b \). Let's use the point \( (4, 220) \):

Substituting into the equation \( y = mx + b \):
\[
220 = 40(4) + b
\]
\[
220 = 160 + b
\]
\[
b = 220 - 160 = 60
\]

### Step 3: Write the Equation

Now we have both the slope \( m \) and the y-intercept \( b \):
\[
y = 40x + 60
\]

Thus, the equation of the linear function that models the relationship between the number of cubic yards of soil ordered and the total cost is:
\[
y = 40x + 60
\]

Answer 5

To compare the rates of change (slopes) of the two linear functions, we need to determine the slopes of each function.

### Linear Function 1:
The equation is given as:
\[
y = x + 8
\]
- The slope \( m_1 \) here is \( 1 \) (the coefficient of \( x \)).

### Linear Function 2:
We need to calculate the slope based on the two given points, \( (2, 6) \) and \( (4, 10) \).

The slope \( m_2 \) can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting in the coordinates:
- \( (x_1, y_1) = (2, 6) \)
- \( (x_2, y_2) = (4, 10) \)

Calculating the slope:
\[
m_2 = \frac{10 - 6}{4 - 2} = \frac{4}{2} = 2
\]

### Comparison:
- The slope of Linear Function 1 (from the equation \( y = x + 8 \)) is \( 1 \).
- The slope of Linear Function 2 (calculated from the points) is \( 2 \).

Since \( 2 > 1 \), Linear Function 2 has the greater rate of change.

### Conclusion:
**Choose 2 for Linear Function 2.**