To find the linear function that models the relationship between the number of cubic yards of soil ordered and the total cost, we can denote the number of cubic yards as \( x \) and the total cost as \( y \).
We're given two points based on Alayah's orders:
- \( (4, 220) \) (4 cubic yards for $220)
- \( (7, 340) \) (7 cubic yards for $340)
Now, let's find the slope \( m \) of the linear function using the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the values from the two points:
- \( (x_1, y_1) = (4, 220) \)
- \( (x_2, y_2) = (7, 340) \)
\[ m = \frac{340 - 220}{7 - 4} = \frac{120}{3} = 40 \]
Now we have the slope \( m = 40 \).
Next, we can use one of the points to find the y-intercept \( b \). We can use the point \( (4, 220) \):
\[ y = mx + b \] \[ 220 = 40(4) + b \] \[ 220 = 160 + b \] \[ b = 220 - 160 = 60 \]
Now we have both \( m \) and \( b \). We can write the equation of the linear function:
\[ y = 40x + 60 \]
Thus, the equation that models the relationship between the number of cubic yards of soil ordered and the total cost is:
\[ \boxed{y = 40x + 60} \]
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