To find the linear equation that models the relationship between the number of cubic yards of soil ordered (x) and the total cost (y), we can use the two sets of data provided to determine the slope (m) and y-intercept (b) of the equation.
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Identify the points:
- From last year: (4, 220)
- From this year: (7, 340)
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Calculate the slope (m): The slope can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the values: \[ m = \frac{340 - 220}{7 - 4} = \frac{120}{3} = 40 \]
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Use point-slope form: We can use one of the points to find the y-intercept (b). We'll use the point (4, 220) and rewrite the line equation as: \[ y = mx + b \] Substituting in the slope and the point: \[ 220 = 40(4) + b \]
This simplifies to: \[ 220 = 160 + b \] Solving for b: \[ b = 220 - 160 = 60 \]
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Write the final equation: The linear equation that models the relationship is: \[ y = 40x + 60 \]
So, the equation in the form \( y = mx + b \) is: \[ y = 40x + 60 \]