Let’s analyze the problem step by step to derive the equations and solutions.
a) Equation for the total cars and trucks for Dealership A:
Let \(x\) be the number of cars and \(y\) be the number of trucks sold by Dealership A. Since the total number of cars and trucks sold by Dealership A is 225, we have:
\[
x + y = 225
\]
Response area a: x + y = 225
b) Equation for the total cars and trucks for Dealership B:
According to the problem:
- Dealership B sells twice as many cars as Dealership A, so the number of cars sold by Dealership B is \(2x\).
- Dealership B sells half as many trucks as Dealership A, so the number of trucks sold by Dealership B is \(\frac{1}{2}y\).
- The total cars and trucks sold by Dealership B is 300.
Thus, the equation for Dealership B can be written as:
\[
2x + \frac{1}{2}y = 300
\]
Response area b: 2x + \frac{1}{2}y = 300
c) How many cars did Dealership A sell?
Now, we have the system of equations:
- \(x + y = 225\)
- \(2x + \frac{1}{2}y = 300\)
First, solve the first equation for \(y\):
\[
y = 225 - x
\]
Now, substitute this expression for \(y\) into the second equation:
\[
2x + \frac{1}{2}(225 - x) = 300
\]
Distribute:
\[
2x + \frac{225}{2} - \frac{1}{2}x = 300
\]
Combine like terms:
\[
\frac{4x - x}{2} + \frac{225}{2} = 300
\]
Multiply through by 2 to eliminate the fraction:
\[
4x - x + 225 = 600
\]
Combine like terms:
\[
3x + 225 = 600
\]
Subtract 225 from both sides:
\[
3x = 375
\]
Divide by 3:
\[
x = 125
\]
So, Dealership A sold 125 cars.
Response area c: 125
d) Given that Dealership B sold half as many trucks as Dealership A, how many trucks did Dealership B sell?
From the equation \(y = 225 - x\), we found \(x = 125\), so:
\[
y = 225 - 125 = 100
\]
Dealership A sold 100 trucks, and since Dealership B sells half as many trucks as Dealership A:
\[
\text{Trucks sold by Dealership B} = \frac{1}{2}(100) = 50
\]
Response area d: 50
Final Answers:
- a: \(x + y = 225\)
- b: \(2x + \frac{1}{2}y = 300\)
- c: 125
- d: 50