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.