Question

Use your graphing calculator: Desmos Graphing CalculatorA company owns two dealerships, both of which sell cars and trucks. Dealership A sells a total of 164 cars and trucks. Dealership B sells twice as many cars and half as many trucks as Dealership A, and sells a total of 229 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) Considering that Dealership B sold half as many trucks as Dealership A, how many trucks did Dealership B sell?

Answer 1

Let's define some variables to help solve the problem:

- Let \( c_A \) be the number of cars sold by Dealership A.
- Let \( t_A \) be the number of trucks sold by Dealership A.
- Let \( c_B \) be the number of cars sold by Dealership B.
- Let \( t_B \) be the number of trucks sold by Dealership B.

### a) Equation for the total cars and trucks for Dealership A:

The total number of vehicles sold by Dealership A (both cars and trucks) can be expressed as:
\[
c_A + t_A = 164
\]

### b) Equation for the total cars and trucks for Dealership B:

According to the problem, Dealership B sells twice as many cars and half as many trucks as Dealership A. Therefore, we can express the number of cars and trucks sold by Dealership B in terms of Dealership A's sales:
- Cars sold by Dealership B: \( c_B = 2c_A \)
- Trucks sold by Dealership B: \( t_B = \frac{1}{2} t_A \)

The total number of vehicles sold by Dealership B is given as:
\[
c_B + t_B = 229
\]

Substituting the expressions for \( c_B \) and \( t_B \):
\[
2c_A + \frac{1}{2}t_A = 229
\]

### c) Solve for how many cars did Dealership A sell:

Now, we have the following system of equations:
1. \( c_A + t_A = 164 \) (Equation 1)
2. \( 2c_A + \frac{1}{2}t_A = 229 \) (Equation 2)

To solve for \( c_A \) and \( t_A \), we can multiply Equation 1 by 2 to eliminate the fraction in Equation 2:
\[
2c_A + 2t_A = 328 \quad \text{(Multiplied by 2)}
\]

Now we can write:

1. \( 2c_A + 2t_A = 328 \)
2. \( 2c_A + \frac{1}{2}t_A = 229 \)

Next, subtract the second equation from the first:
\[
(2c_A + 2t_A) - (2c_A + \frac{1}{2}t_A) = 328 - 229
\]

This simplifies to:
\[
2t_A - \frac{1}{2}t_A = 99
\]
\[
\frac{4}{2}t_A - \frac{1}{2}t_A = 99
\]
\[
\frac{3}{2}t_A = 99
\]

Now, multiply both sides by \( \frac{2}{3} \):
\[
t_A = \frac{2}{3} \times 99 = 66
\]

Now substitute \( t_A \) back into Equation 1 to find \( c_A \):
\[
c_A + 66 = 164
\]
\[
c_A = 164 - 66 = 98
\]

So, Dealership A sold:
**c)** \( c_A = 98 \) cars.

### d) Trucks sold by Dealership B:

Since Dealership B sells half as many trucks as Dealership A:
\[
t_B = \frac{1}{2} t_A = \frac{1}{2} \times 66 = 33
\]

So, Dealership B sold:
**d)** \( t_B = 33 \) trucks.

### Summary of Answers:
a) \( c_A + t_A = 164 \)
b) \( 2c_A + \frac{1}{2}t_A = 229 \)
c) Dealership A sold \( 98 \) cars.
d) Dealership B sold \( 33 \) trucks.