Let's break this down step by step.
a) Equation for Dealership A
You've already provided the correct equation for Dealership A: \[ x + y = 164 \] where \( x \) represents the number of cars sold, and \( y \) represents the number of trucks sold.
b) Equation for Dealership B
According to the problem, Dealership B sells twice as many cars and half as many trucks as Dealership A. If we let \( x \) be the number of cars sold by Dealership A and \( y \) be the number of trucks sold by Dealership A, then:
- The number of cars sold by Dealership B is \( 2x \).
- The number of trucks sold by Dealership B is \( \frac{1}{2}y \).
The total number of cars and trucks sold by Dealership B is given to be 229: \[ 2x + \frac{1}{2}y = 229 \]
c) How many cars did Dealership A sell?
Now we have the following system of equations:
- \( x + y = 164 \)
- \( 2x + \frac{1}{2}y = 229 \)
We can solve these equations simultaneously. First, we can solve the first equation for \( y \): \[ y = 164 - x \]
Now we'll substitute \( y \) into the second equation: \[ 2x + \frac{1}{2}(164 - x) = 229 \]
Distributing: \[ 2x + 82 - \frac{1}{2}x = 229 \]
Combining like terms: \[ \frac{4x - x}{2} + 82 = 229 \] \[ \frac{3x}{2} + 82 = 229 \]
Subtract 82 from both sides: \[ \frac{3x}{2} = 147 \]
Multiply both sides by \( \frac{2}{3} \): \[ x = 98 \]
d) How many trucks did Dealership B sell?
Now that we know that Dealership A sold \( x = 98 \) cars, let's find out how many trucks were sold by Dealership A using the equation \( y = 164 - x \): \[ y = 164 - 98 = 66 \]
Now we can find the number of trucks sold by Dealership B, which is half the number of trucks sold by Dealership A: \[ \text{Trucks sold by Dealership B} = \frac{1}{2}y = \frac{1}{2} \times 66 = 33 \]
Summary of Answers:
a) Equation for Dealership A: \( x + y = 164 \)
b) Equation for Dealership B: \( 2x + \frac{1}{2}y = 229 \)
c) Dealership A sold 98 cars.
d) Dealership B sold 33 trucks.