To solve this problem, we'll first define \( x \) as the number of cars sold at Dealership A and \( y \) as the number of trucks sold at Dealership A.
a) An equation for the total cars and trucks for Dealership A:
Since Dealership A sells a total of 164 cars and trucks, the equation is:
- Response Area 1: \( x + y = 164 \)
b) An equation for the total cars and trucks for Dealership B:
Dealership B sells twice as many cars as Dealership A could be expressed as \( 2x \) and half as many trucks as Dealership A could be expressed as \( \frac{1}{2}y \). Since the total sales at Dealership B is 229 cars and trucks, the equation is:
- Response Area 2: \( 2x + \frac{1}{2}y = 229 \)
c) How many cars did Dealership A sell?
We can solve the system of equations:
- \( x + y = 164 \) (Equation for Dealership A)
- \( 2x + \frac{1}{2}y = 229 \) (Equation for Dealership B)
From the first equation, we can express \( y \) in terms of \( x \):
Now substitute \( y \) into the second equation:
- \( 2x + \frac{1}{2}(164 - x) = 229 \)
- Simplifying gives:
\( 2x + 82 - \frac{1}{2}x = 229 \)
\( \frac{4x - x}{2} + 82 = 229 \)
\( \frac{3x}{2} + 82 = 229 \)
\( \frac{3x}{2} = 229 - 82 \)
\( \frac{3x}{2} = 147 \)
\( 3x = 294 \)
\( x = 98 \)
So Dealership A sold:
- Response Area 3: \( 98 \)
d) Considering that Dealership B sold half as many trucks as Dealership A, how many trucks did Dealership B sell?
Since Dealership B sells half the number of trucks as Dealership A, we will first find the number of trucks sold by Dealership A:
- From \( y = 164 - x \)
- \( y = 164 - 98 = 66 \) (trucks sold at Dealership A)
Dealership B sells half of this:
- Trucks sold by Dealership B: \( \frac{1}{2}y = \frac{1}{2}(66) = 33 \)
So, Dealership B sold:
- Response Area 4: \( 33 \)
Summary of Responses:
- Response Area 1: \( x + y = 164 \)
- Response Area 2: \( 2x + \frac{1}{2}y = 229 \)
- Response Area 3: \( 98 \) (cars sold by Dealership A)
- Response Area 4: \( 33 \) (trucks sold by Dealership B)
