Use your graphing calculator: Desmos Graphing Calculator 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: b) An equation for the total cars and trucks for Dealership B: c) How many cars did Dealership A sell? d) Given the fact that Dealership B sold half as many trucks as Dealership A, how many trucks did Dealership B sell?
a) An equation for the total cars and trucks for Dealership A:
Let x be the number of cars sold by Dealership A and y be the number of trucks sold by Dealership A.
The total cars and trucks sold by Dealership A can be represented by the equation: x + y = 225.
b) An equation for the total cars and trucks for Dealership B:
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 (1/2)y.
The total cars and trucks sold by Dealership B can be represented by the equation: 2x + (1/2)y = 300.
c) To find the number of cars sold by Dealership A, we can substitute the value of y from equation a) into equation b) and solve for x:
2x + (1/2)y = 300
2x + (1/2)(225 - x) = 300
2x + 112.5 - (1/2)x = 300
(3/2)x + 112.5 = 300
(3/2)x = 300 - 112.5
(3/2)x = 187.5
x = (2/3) * 187.5
x ≈ 125
Dealership A sold approximately 125 cars.
d) Given that Dealership B sold half as many trucks as Dealership A, we can substitute the value of x from part c) into equation a) and solve for y:
x + y = 225
125 + y = 225
y = 225 - 125
y = 100
Dealership B sold 100 trucks.
a) To find the equation for the total cars and trucks for Dealership A, we can define the variables: let C denote the number of cars and T denote the number of trucks sold by Dealership A.
The equation for the total cars and trucks for Dealership A can be written as:
C + T = 225
b) To find the equation for the total cars and trucks for Dealership B, we use the information given: Dealership B sells twice as many cars and half as many trucks as Dealership A. Using the variables defined earlier, the equation for Dealership B can be written as:
2C + (1/2)T = 300
c) To determine how many cars Dealership A sold, we can solve the equation C + T = 225 for C. Rearranging the equation, we get:
C = 225 - T
d) Given the fact that Dealership B sold half as many trucks as Dealership A, we can set the number of trucks sold by Dealership B equal to half the number of trucks sold by Dealership A. Using the variables defined earlier, we can write the equation:
T_B = (1/2)T_A
To find the number of trucks sold by Dealership B, we need to solve for T_B. We know the total number of cars and trucks sold by Dealership B is 300 (from part b), so we substitute the values into the equation we just wrote:
2C_B + (1/2)T_B = 300
Now, we can substitute C in terms of T from part c:
2(225 - T_B) + (1/2)T_B = 300
Solving this equation will give us the number of trucks sold by Dealership B.
a) An equation for the total cars and trucks for Dealership A:
Let's assume the number of cars sold by Dealership A is C and the number of trucks sold is T. According to the problem statement, the total number of cars and trucks sold by Dealership A is 225. So, the equation for Dealership A can be written as:
C + T = 225
b) An equation for the total cars and trucks for Dealership B:
The problem states that Dealership B sells twice as many cars as Dealership A and half as many trucks as Dealership A. Let's represent the number of cars sold by Dealership B as C' and the number of trucks sold as T'. According to the problem statement, the total number of cars and trucks sold by Dealership B is 300. So, we can write the equation for Dealership B as:
C' + T' = 300
c) To find out how many cars Dealership A sold, we can substitute the value of T from equation a) into equation b):
C + (2C)/2 = 300
Simplifying this equation, we get:
C + C = 300
2C = 300
Dividing both sides by 2, we find:
C = 150
Therefore, Dealership A sold 150 cars.
d) Given that Dealership B sold half as many trucks as Dealership A, we can substitute the value of C and T from equation a) into equation b):
150 + T' = 300
Subtracting 150 from both sides of the equation, we get:
T' = 150
Therefore, Dealership B sold 150 trucks.