To find 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 start by determining the slope (m) using the given points.
We have two points:
- (4, 220): 4 cubic yards cost $220.
- (7, 340): 7 cubic yards cost $340.
The formula for the slope \( m \) of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the values from our points:
\[ m = \frac{340 - 220}{7 - 4} = \frac{120}{3} = 40 \]
Therefore, the slope \( m = 40 \).
Now we can use one of the points to find the y-intercept \( b \). We can use the point (4, 220) and the slope we just calculated:
Using the equation \( y = mx + b \):
\[ 220 = 40(4) + b \] \[ 220 = 160 + b \] \[ b = 220 - 160 = 60 \]
Now we have both the slope \( m \) and the y-intercept \( b \).
The equation of the linear function is:
\[ y = 40x + 60 \]
This represents the relationship between the number of cubic yards of soil ordered (x) and the total cost (y).