To find the equation of the linear function in the form \( y = mx + b \), where \( y \) is the total cost, \( x \) is the number of cubic yards of soil ordered, \( m \) is the slope of the line, and \( b \) is the y-intercept, we can start by identifying the points based on Elena's orders.
From the information given:
- Last year: 3 cubic yards for $90. This gives us the point (3, 90).
- This year: 6 cubic yards for $150. This gives us the point (6, 150).
Step 1: Calculate the slope (m)
The slope \( m \) of the line can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points (3, 90) and (6, 150):
- \( (x_1, y_1) = (3, 90) \)
- \( (x_2, y_2) = (6, 150) \)
Now, substitute into the formula: \[ m = \frac{150 - 90}{6 - 3} = \frac{60}{3} = 20 \]
Step 2: Find the y-intercept (b)
We can use one of the points to solve for \( b \). We will use the point (3, 90) and the slope we just calculated.
The equation of the line is: \[ y = mx + b \] Substituting \( m = 20 \): \[ 90 = 20(3) + b \] \[ 90 = 60 + b \] \[ b = 90 - 60 = 30 \]
Final Equation
Now that we have \( m \) and \( b \), we can write the equation: \[ y = 20x + 30 \]
Thus, the linear function that models the relationship between the number of cubic yards of soil ordered and the total cost is: \[ \boxed{y = 20x + 30} \]