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 follow these steps:
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Identify the two points: We have two data points based on Alayah's orders:
- Last year: \( (4, 220) \)
- This year: \( (7, 340) \)
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Calculate the slope (m): The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Plugging in the points: \[ m = \frac{340 - 220}{7 - 4} = \frac{120}{3} = 40 \]
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Use one point to find the y-intercept (b): We'll use the point \( (4, 220) \) and the slope \( m = 40 \) in the line equation \( y = mx + b \): \[ 220 = 40(4) + b \] \[ 220 = 160 + b \] \[ b = 220 - 160 = 60 \]
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Write the linear equation: Now we can substitute \( m \) and \( b \) into the linear equation: \[ y = 40x + 60 \]
So the final equation of the linear function that models the relationship between the number of cubic yards of soil ordered and the total cost is:
\[ y = 40x + 60 \]