Jonathan purchased 30 bags of mulch to spread in his garden. The graph shows points representing the number of bags of mulch left to spread at several different times while Jonathan is working.

Question

Select from the drop-down menus to complete the equation, in standard form, that represents the relationship between the number of bags of mulch left to spread, y, and the time, x, Jonathan has spent mulching.

_____x + y = _____

Bags of much left, are on the left and the numbers are 5 10 15 20 25 30 35 and on the bottom are hours and the number go from 1 2 3 4. And the dots on the lines are 2 (2,12), 1 (1,21), and 0 (0.30)

Answer 1

To create the equation in standard form that represents the relationship between the number of bags of mulch left to spread (y) and the time (x) spent mulching, we first need to identify the linear relationship from the points given.

The points provided are:

• (0, 30)
• (1, 21)
• (2, 12)

From these points, we can determine the slope (m) of the line. The slope is defined as the change in y divided by the change in x.

Using the first two points:

• Point 1: (0, 30) → y-intercept (at x = 0, y = 30)
• Point 2: (1, 21)

The change in y (Δy) from 30 to 21 is: \[ \Delta y = 21 - 30 = -9 \] The change in x (Δx) from 0 to 1 is: \[ \Delta x = 1 - 0 = 1 \]

Thus, the slope (m) can be calculated as: \[ m = \frac{\Delta y}{\Delta x} = \frac{-9}{1} = -9 \]

Now, we can use the point-slope formula to determine the equation of the line. We can use the y-intercept (0, 30): \[ y - 30 = -9(x - 0) \] This simplifies to: \[ y - 30 = -9x \] Adding 30 to both sides gives: \[ y = -9x + 30 \]

Now, to convert this into standard form \( Ax + By = C \): Adding \( 9x \) to both sides results in: \[ 9x + y = 30 \]

To complete the equation in standard form: The final equation is therefore: \[ 9x + y = 30 \]

So the completed equation is: 9x + y = 30.