The equation

The equation \( y = mx + b \) defines:

a. a linear function

This equation represents a straight line in a Cartesian coordinate system, where \( m \) is the slope and \( b \) is the y-intercept.

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.

To find the equation of the linear function in the form \( y = mx + b \), we can follow these steps:

1. **Calculate the slope \( m \)** using the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points \( (6, 16) \) and \( (15, 43) \):

\[
m = \frac{43 - 16}{15 - 6} = \frac{27}{9} = 3
\]

2. **Use the slope-intercept form** of the equation \( y = mx + b \) and substitute one of the points to find \( b \). Let's use the point \( (6, 16) \):

\[
16 = 3(6) + b
\]
\[
16 = 18 + b
\]
\[
b = 16 - 18 = -2
\]

3. **Write the equation** using the slope \( m \) and the y-intercept \( b \):

\[
y = 3x - 2
\]

Thus, the equation of the linear function that models the relationship shown in the table is:

\[
y = 3x - 2
\]

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.

To write the equation of the linear function that models the relationship between the number of cubic yards of soil ordered (x) and the total cost (y), we can follow these steps:

1. **Identify the points** representing the data given:
- Last year: \( (4, 220) \)
- This year: \( (7, 340) \)

2. **Calculate the slope \( m \)** using the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points \( (4, 220) \) and \( (7, 340) \):

\[
m = \frac{340 - 220}{7 - 4} = \frac{120}{3} = 40
\]

3. **Use the slope-intercept form** of the equation \( y = mx + b \) to find \( b \). We can use either point; let's use the point \( (4, 220) \):

\[
220 = 40(4) + b
\]
\[
220 = 160 + b
\]
\[
b = 220 - 160 = 60
\]

4. **Write the equation** using 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:

\[
y = 40x + 60
\]

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: ( 4, 10) (2, 6)

Linear Function ___ has the greater rate of change.

To compare the rates of change of the two linear functions, we need to examine their slopes.

**Linear Function 1:**
The equation is given as \( y = x + 8 \). The slope (\( m \)) of this linear function is **1**.

**Linear Function 2:**
We need to calculate the slope from the two given points \( (4, 10) \) and \( (2, 6) \). The formula for the slope between two points is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points:
- \( (x_1, y_1) = (4, 10) \)
- \( (x_2, y_2) = (2, 6) \)

Now substitute the values into the formula:

\[
m = \frac{6 - 10}{2 - 4} = \frac{-4}{-2} = 2
\]

Now we can compare the slopes:
- Linear Function 1 has a slope of **1**.
- Linear Function 2 has a slope of **2**.

Since **2 > 1**, this means that:

**Linear Function 2 has the greater rate of change.**

Thus, the final answer is:

Linear Function **2** has the greater rate of change.