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A 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: x + y = 225 b) An equation for the total cars and trucks for Dealership B: 2x + 1/2y = 300 î c) How many cars did Dealership A sell? 125 d) Given the fact that Dealership B sold half as many trucks as Dealership A, how many trucks did Dealership B sell?
c) Dealership A sold 125 cars.
To find this, we solve the equation x + y = 225, where x represents the number of cars and y represents the number of trucks. Since we are asked for the number of cars, we need to find x. We can rearrange the equation to solve for x:
x = 225 - y
Since we don't know the exact value of y, we can plug in different values for y and solve for x. However, we are given the information in part d) that Dealership B sold half as many trucks as Dealership A. So the number of trucks sold by Dealership B is y/2.
Substituting this into the equation 2x + (y/2) = 300, we get:
2x + y/2 = 300
Now we can solve for x:
2x = 300 - y/2
x = (300 - y/2) / 2
Substituting the value of x from equation (1):
125 = (300 - y/2) / 2
To solve for y, we need to multiply both sides of the equation by 2:
250 = 300 - y/2
y/2 = 300 - 250
y/2 = 50
Multiplying both sides of the equation by 2:
y = 100
So, Dealership B sold 100 trucks.
To solve this problem, let's analyze the information given:
Dealership A sells a total of 225 cars and trucks.
Dealership B sells twice as many cars and half as many trucks as Dealership A, and sells a total of 300 cars and trucks.
a) An equation for the total cars and trucks for Dealership A: x + y = 225
This equation represents the total number of cars (x) and trucks (y) that Dealership A sells, equal to 225.
b) An equation for the total cars and trucks for Dealership B: 2x + 1/2y = 300
This equation represents the total number of cars (2x) and trucks (1/2y) that Dealership B sells, equal to 300.
c) To find out how many cars Dealership A sold, we can set up an equation by using the information from equation (a):
x + y = 225
We know that x represents the number of cars Dealership A sold, so we can rewrite the equation as:
x + y = 225
x + y = 300
Subtracting equation (b) from equation (a):
x + y = 225
- (2x + 1/2y) = 300
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-2x - y = -75
Now we have a system of equations:
x + y = 225
-2x - y = -75
To solve this system, we can either use substitution or elimination method. Let's use the elimination method:
Multiply equation (a) by -1:
-1(x + y) = -1(225)
-1x - y = -225
Add this new equation to equation (b):
-2x - y + -1x - y = -75 + (-225)
-3x - 2y = -300
Now we have the following equation:
-3x - 2y = -300
Let's multiply equation (a) by 2:
2(x + y) = 2(225)
2x + 2y = 450
Add this new equation to equation (b):
2x + 2y + -2x - y = 450 + (-75)
y = 375
Now that we have the value for y, we can substitute it back into equation (a):
x + y = 225
x + 375 = 225
x = 225 - 375
x = -150
However, since it doesn't make sense for a dealership to sell a negative number of cars, it seems like there might be an error in the information given. Please double-check the values provided in the problem statement.
To find the number of cars Dealership A sold, we can use the equation x + y = 225 where x represents the number of cars and y represents the number of trucks. Since we want to find the number of cars, we can isolate x in the equation.
x + y = 225
x = 225 - y
From the information provided, we know that Dealership B sells twice as many cars as Dealership A. Therefore, we can say that the equation for the total cars and trucks for Dealership B is 2x + 1/2y = 300. Using the value of x from the equation we derived for Dealership A, we substitute it into the equation for Dealership B.
2x + 1/2y = 300
2(225 - y) + 1/2y = 300
Simplifying the equation:
450 - 2y + 1/2y = 300
-2y + 1/2y = 300 - 450
-3/2y = -150
y = (-150) / (-3/2)
y = (-150) x (-2/3)
y = 100
Therefore, Dealership A sold 125 cars (x = 225 - y = 225 - 100 = 125). And given that Dealership B sold half as many trucks as Dealership A, Dealership B sold 50 trucks.