Let's break down the problem step by step.
To solve for the number of cars and trucks sold by both dealerships, we can define:
- \( x \): the number of cars sold at Dealership A
- \( y \): the number of trucks sold at Dealership A
a) An equation for the total cars and trucks for Dealership A:
The total number of vehicles sold at Dealership A is the sum of cars and trucks:
\[ x + y = 164 \]
b) An equation for the total cars and trucks for Dealership B:
According to the problem, Dealership B sells twice as many cars as Dealership A and half as many trucks as Dealership A. Therefore:
- The number of cars sold at Dealership B is \( 2x \)
- The number of trucks sold at Dealership B is \( \frac{y}{2} \)
The total number of vehicles sold at Dealership B is then:
\[ 2x + \frac{y}{2} = 229 \]
c) Solve the equations:
Now we have a system of equations:
- \( x + y = 164 \) (1)
- \( 2x + \frac{y}{2} = 229 \) (2)
From equation (1), we can express \( y \) in terms of \( x \):
\[ y = 164 - x \]
Substituting this value of \( y \) into equation (2):
\[ 2x + \frac{164 - x}{2} = 229 \]
Now, to eliminate the fraction, multiply every term by 2:
\[ 4x + 164 - x = 458 \]
Combine like terms:
\[ 3x + 164 = 458 \]
Subtract 164 from both sides:
\[ 3x = 294 \]
Divide by 3:
\[ x = 98 \]
So, Dealership A sold 98 cars.
d) Now, how many trucks did Dealership A and Dealership B sell:
Using the value of \( x \) in the equation for \( y \):
\[ y = 164 - x = 164 - 98 = 66 \]
So, Dealership A sold 66 trucks.
Now, for Dealership B:
- The number of trucks sold at Dealership B is \( \frac{y}{2} = \frac{66}{2} = 33 \).
So, Dealership B sold 33 trucks.
Final Answers:
- c) Dealership A sold 98 cars.
- d) Dealership B sold 33 trucks.
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