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:
\[ y = 40x + 60 \]
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