Question

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

Answer 1

To find the equation of the linear function that models the relationship between the number of cubic yards of soil ordered (let \( x \) be the number of cubic yards) and the total cost (let \( y \) be the total cost), we can use the two points provided from Alayah's orders.

From the information provided:
- Last year: 4 cubic yards for $220 gives us the point (4, 220).
- This year: 7 cubic yards for $340 gives us the point (7, 340).

We can find the slope \( m \) of the linear equation using the formula:

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

Where:
- \( (x_1, y_1) = (4, 220) \)
- \( (x_2, y_2) = (7, 340) \)

Substituting in the values:

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

Now that we have the slope \( m = 40 \), we can use one of the points to find the y-intercept \( b \) using the equation of the line \( y = mx + b \).

Using the point (4, 220):

\[
220 = 40(4) + b
\]

Calculating:

\[
220 = 160 + b
\]

Solving for \( b \):

\[
b = 220 - 160 = 60
\]

Now we have the slope and the y-intercept. The equation of the linear function is:

\[
y = 40x + 60
\]

Thus, the final answer is:

\[
y = 40x + 60
\]