Question

Use your graphing calculator: Desmos Graphing Calculator

A company owns two dealerships, both of which sell cars and trucks. Dealership A sells a total of 225 cars and trucks. Dealership B sells twice as many cars and half as many trucks as the Dealership A, and sells a total of 300 cars and trucks.

(4 points)
a) An equation for the total cars and trucks for Dealership A:

b) An equation for the total cars and trucks for Dealership B:

c) How many cars did Dealership A sell?

d) Given the fact that Dealership B sold half as many trucks as Dealership A, how many trucks did Dealership B sell?

Skip to navigation



















x +y=225
x+y=525
x+y=300

1/2x +2y = 225
1/2x + 1/2y = 525
2x + 1/2y = 300

132
125
98
196

49
33
50
98

Answer 1

Let's denote:

- \( x_A \): the number of cars sold at Dealership A
- \( y_A \): the number of trucks sold at Dealership A
- \( x_B \): the number of cars sold at Dealership B
- \( y_B \): the number of trucks sold at Dealership B

Based on the problem information, we have the following information:

1. Dealership A sells a total of 225 cars and trucks:
\[
x_A + y_A = 225
\]

2. Dealership B sells twice as many cars and half as many trucks as Dealership A, and a total of 300 cars and trucks:
- Cars: \( x_B = 2x_A \)
- Trucks: \( y_B = \frac{1}{2} y_A \)

Therefore, for Dealership B:
\[
x_B + y_B = 300
\]

Substituting \( x_B \) and \( y_B \) into the total for Dealership B:
\[
2x_A + \frac{1}{2}y_A = 300
\]

### (a) Equation for the total cars and trucks for Dealership A:
\[
x_A + y_A = 225
\]

### (b) Equation for the total cars and trucks for Dealership B:
Substituting for Dealership B's cars and trucks into the total:
\[
2x_A + \frac{1}{2}y_A = 300
\]

### (c) To find how many cars Dealership A sold, we can solve these two equations:
1. \( x_A + y_A = 225 \) (1)
2. \( 2x_A + \frac{1}{2}y_A = 300 \) (2)

First, we can rearrange equation (1):
\[
y_A = 225 - x_A
\]

Substituting \( y_A \) into equation (2):
\[
2x_A + \frac{1}{2}(225 - x_A) = 300
\]

Now, simplify the equation:
\[
2x_A + \frac{225}{2} - \frac{1}{2}x_A = 300
\]
\[
\frac{4x_A - x_A + 225}{2} = 300
\]
\[
\frac{3x_A + 225}{2} = 300
\]

Multiply both sides by 2:
\[
3x_A + 225 = 600
\]

Now, subtract 225 from both sides:
\[
3x_A = 375
\]

Divide by 3:
\[
x_A = 125
\]

So, the number of cars Dealership A sold is:
\[
\boxed{125}
\]

### (d) Now we can find how many trucks Dealership A sold and then Dealership B's trucks:
Substituting \( x_A = 125 \) back into equation (1):
\[
125 + y_A = 225
\]
\[
y_A = 100
\]

Now, using the relationship \( y_B = \frac{1}{2}y_A \):
\[
y_B = \frac{1}{2} \times 100 = 50
\]

Therefore, the number of trucks Dealership B sold is:
\[
\boxed{50}
\]