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Last year, Alayah ordered 4 cubic yards of soil and spent $220. This year, she orders 7 cubic yard of soil and spends $340. Write the equation of the linear function that models the relationship between the number of cubic yards of soil ordered and the total cost. Enter your answer in y = mx + b form

3 answers

To determine 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 data points provided:

  1. When Alayah ordered 4 cubic yards (x = 4), the cost was $220 (y = 220).
  2. When Alayah ordered 7 cubic yards (x = 7), the cost was $340 (y = 340).

We can represent this information as two points: \((4, 220)\) and \((7, 340)\).

To find the slope (m) of the line, we use the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Using the points (4, 220) and (7, 340):

\[ m = \frac{340 - 220}{7 - 4} = \frac{120}{3} = 40 \]

Now that we have the slope (m = 40), we can use one of the points to find the y-intercept (b). We'll use the point (4, 220):

Using the slope-intercept form \(y = mx + b\):

\[ 220 = 40(4) + b \] \[ 220 = 160 + b \] \[ b = 220 - 160 = 60 \]

Now we have both the slope and the y-intercept. The equation of the linear function is:

\[ y = 40x + 60 \]

Thus, the linear equation that models the relationship is:

\[ y = 40x + 60 \]

you sure

Let’s double-check the calculations carefully to ensure the equation is correct.

  1. We have two points: \((4, 220)\) and \((7, 340)\).

  2. To calculate the slope \( m \):

    \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{340 - 220}{7 - 4} = \frac{120}{3} = 40 \]

    So, the slope \( m = 40\).

  3. Now we can use this slope and one of the points to find the y-intercept \( b \). Let’s use the point \((4, 220)\):

    The slope-intercept form is given by:

    \[ y = mx + b \]

    Substituting in the point \( (4, 220) \):

    \[ 220 = 40(4) + b \]

    Calculating \( 40(4) \):

    \[ 220 = 160 + b \]

    Now, subtract \( 160 \) from both sides:

    \[ b = 220 - 160 = 60 \]

  4. Now, with \( m = 40 \) and \( b = 60 \), we write the equation:

    \[ y = 40x + 60 \]

So indeed, the equation \( y = 40x + 60 \) is correct based on the calculations.

If Alayah orders \( x \) cubic yards, the total cost \( y \) can be accurately modeled by that equation.

If you have any further modifications or if you meant something else, please let me know!