Use your graphing calculator: Desmos Graphing CalculatorA 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?

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 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?

GPT-4o mini · Answer

Let's define the variables for the number of cars and trucks sold at Dealership A and Dealership B:

Let \( c_A \) = number of cars sold at Dealership A
Let \( t_A \) = number of trucks sold at Dealership A
Let \( c_B \) = number of cars sold at Dealership B
Let \( t_B \) = number of trucks sold at Dealership B

Now, we can break down the problem step by step:

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

Given that Dealership A sells a total of 225 cars and trucks, we write the equation as: \[ c_A + t_A = 225 \]

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. Based on this, the equations for cars and trucks sold at Dealership B can be expressed as: \[ c_B = 2c_A \] \[ t_B = \frac{1}{2}t_A \] Additionally, we know that Dealership B sells a total of 300 cars and trucks: \[ c_B + t_B = 300 \]

Replace \( c_B \) and \( t_B \) in the total equation for Dealership B:

Substituting \( c_B \) and \( t_B \) into the equation gives us: \[ 2c_A + \frac{1}{2}t_A = 300 \]

Now we have the following system of equations:

\( c_A + t_A = 225 \)
\( 2c_A + \frac{1}{2}t_A = 300 \)

c) How many cars did Dealership A sell?

We can solve the system of equations to find the number of cars sold at Dealership A.

From the first equation, we can express \( t_A \): \[ t_A = 225 - c_A \]

Substituting this expression for \( t_A \) in the second equation: \[ 2c_A + \frac{1}{2}(225 - c_A) = 300 \] Simplifying this: \[ 2c_A + 112.5 - \frac{1}{2}c_A = 300 \] Combining terms: \[ \frac{3}{2}c_A + 112.5 = 300 \] Subtracting 112.5 from both sides: \[ \frac{3}{2}c_A = 187.5 \] Multiplying both sides by \( \frac{2}{3} \): \[ c_A = 125 \]

Answer for c): Dealership A sold 125 cars.

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

First, we can find \( t_A \): \[ t_A = 225 - c_A = 225 - 125 = 100 \]

Since Dealership B sells half as many trucks as Dealership A, we have: \[ t_B = \frac{1}{2}t_A = \frac{1}{2}(100) = 50 \]

Answer for d): Dealership B sold 50 trucks.

Summary:

a) \( c_A + t_A = 225 \)
b) \( 2c_A + \frac{1}{2}t_A = 300 \)
c) 125 cars sold by Dealership A.
d) 50 trucks sold by Dealership B.