Morgan owns a car dealership in which her employees earn commission for each car they sell in addition to a weekly salary. One employee sells 5 cars and makes $1,875 that week. A second employee sells 6 cars and makes $2,150 that week.

Question

Enter an equation, in slope-intercept form, for the amount of money  y
  an employee will make in a week given the number of cars sold  x
 .

GPT-4o mini · Answer

To find the equation in slope-intercept form $ y = mx + b $, where $ y $ represents the total earnings, $ x $ represents the number of cars sold, $ m $ is the slope (the commission earned per car sold), and $ b $ is the weekly salary.

Let's denote:

$ y_1 = 1875 $ when $ x_1 = 5 $ (for the first employee)
$ y_2 = 2150 $ when $ x_2 = 6 $ (for the second employee)

We can set up a system of equations based on the given information:

$ 1,875 = m \cdot 5 + b $
$ 2,150 = m \cdot 6 + b $

To solve these equations, we can start by eliminating $ b $:

From the first equation: \[ b = 1875 - 5m \]

Now, substitute $ b $ into the second equation: \[ 2150 = 6m + (1875 - 5m) \]

This simplifies to: \[ 2150 = 6m + 1875 - 5m \] \[ 2150 = m + 1875 \] \[ m = 2150 - 1875 = 275 \]

Now that we have the value of $ m $, we can substitute it back to find $ b $: \[ b = 1875 - 5 \cdot 275 \] \[ b = 1875 - 1375 = 500 \]

Now, we have $ m = 275 $ and $ b = 500 $. The equation in slope-intercept form is: \[ y = 275x + 500 \]

Therefore, the equation representing the earnings of an employee based on the number of cars sold is: \[ \boxed{y = 275x + 500} \]

NICKOLAS · Answer

How much will an employee earn for selling 10 cars in one week? Enter the answer.

GPT-4o mini · Answer

To find out how much an employee will earn for selling 10 cars in one week, we can use the equation we derived earlier:

\[ y = 275x + 500 \]

where $ x $ is the number of cars sold. We will substitute $ x = 10 $ into the equation:

\[ y = 275(10) + 500 \] \[ y = 2750 + 500 \] \[ y = 3250 \]

Therefore, an employee will earn $3,250 for selling 10 cars in one week.