To write a linear equation based on Aaliya's soil orders, we can first express the relationship between the quantity of soil ordered (in cubic yards) and the cost (in dollars).
From the information given:
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Last year:
- Quantity (x) = 4 cubic yards
- Cost (y) = $220
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This year:
- Quantity (x) = 7 cubic yards
- Cost (y) = $340
We can find the slope (m) of the linear equation using the two points (4, 220) and (7, 340) using the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{340 - 220}{7 - 4} = \frac{120}{3} = 40 \]
So the slope \( m \) is 40. This means that for each additional cubic yard of soil, the cost increases by $40.
Next, we can write the equation in point-slope form or slope-intercept form. Using the point (4, 220), we can use the point-slope form:
\[ y - y_1 = m(x - x_1) \]
Substituting in the values we have:
\[ y - 220 = 40(x - 4) \]
Now, simplifying this equation to slope-intercept form \( y = mx + b \):
\[ y - 220 = 40x - 160 \] \[ y = 40x + 60 \]
So, the linear equation that represents the relationship between the quantity of soil ordered and the cost is:
\[ y = 40x + 60 \]
Where:
- \( y \) is the cost in dollars,
- \( x \) is the quantity of soil in cubic yards.
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