Question

The past 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 1

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

The two data points we have are:
1. When \( x = 4 \), \( y = 220 \)
2. When \( x = 7 \), \( y = 340 \)

We can use these points to find the slope \( m \) of the line using the formula:

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

Substituting in our points \((x_1, y_1) = (4, 220)\) and \((x_2, y_2) = (7, 340)\):

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

Now that we have the slope \( m = 40 \), we can use the point-slope form of the linear equation to find the y-intercept \( b \). We'll use one of the points, let's use \( (4, 220) \):

The point-slope form of the equation is given by:

\[
y - y_1 = m(x - x_1)
\]

Substituting \( m = 40 \), \( x_1 = 4 \), and \( y_1 = 220 \):

\[
y - 220 = 40(x - 4)
\]

Now simplify this equation:

\[
y - 220 = 40x - 160
\]

Add 220 to both sides:

\[
y = 40x + 60
\]

So, the equation that models the relationship between the number of cubic yards of soil ordered and the total cost is:

\[
y = 40x + 60
\]