To derive the linear function that models the relationship between the number of cubic yards of soil (let's denote this as \(x\)) and the total cost (let's denote this as \(y\)), we can use the two given points based on Alayah's purchases:
- The first point from last year: (4 cubic yards, $220) → \((4, 220)\)
- The second point from this year: (7 cubic yards, $340) → \((7, 340)\)
To find the linear function \(y = mx + b\), we first need to calculate the slope \(m\) using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the values from our points:
\[ m = \frac{340 - 220}{7 - 4} = \frac{120}{3} = 40 \]
Now that we have the slope \(m = 40\), we can use one of the points to find the y-intercept \(b\). Let's use the point (4, 220):
The equation can be set up as follows:
\[ y = mx + b \] \[ 220 = 40(4) + b \] \[ 220 = 160 + b \] \[ b = 220 - 160 = 60 \]
Now we have both the slope and the y-intercept. The linear equation that models the relationship is:
\[ y = 40x + 60 \]
Thus, the equation of the linear function is:
\[ y = 40x + 60 \]
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