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A company owns two dealerships, both of which sell cars and trucks. Dealership A sells a total of 164 cars and trucks. Dealership B sells twice as many cars and half as many trucks as Dealership A, and sells a total of 229 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) Considering that Dealership B sold half as many trucks as Dealership A, how many trucks did Dealership B sell?
a) Let's use the variables "ca" and "ta" to represent the number of cars and trucks sold by Dealership A, respectively. The equation for the total cars and trucks for Dealership A can be written as:
ca + ta = 164
b) Using the same logic, let's use the variables "cb" and "tb" to represent the number of cars and trucks sold by Dealership B, respectively. The equation for the total cars and trucks for Dealership B can be written as:
cb + tb = 229
c) From the given information, we know that Dealership A sells a total of 164 cars and trucks. We can use the equation we defined in part a) to solve for the number of cars (ca) sold by Dealership A:
ca + ta = 164
ca + tb = 29
Substituting the value of tb from equation b) into equation a):
ca + (229 - cb) = 164
ca = 164 - 229 + cb
ca = cb - 65
d) Considering that Dealership B sold half as many trucks as Dealership A, we can write the equation:
tb = 0.5 * ta
We can substitute this equation into equation b):
cb + 0.5 * ta = 229
Substituting the value of ta from equation a) into equation b):
cb + 0.5 * (ca + tb) = 229
cb + 0.5 * (cb - 65 + tb) = 229
Simplifying the equation:
1.5 * cb + 0.5 * tb = 229
1.5 * cb + 0.5 * (229 - cb) = 229
Simplifying further:
1.5 * cb + 0.5 * 229 - 0.5 * cb = 229
cb + 0.5 * 229 = 229
Solving this equation gives us cb = 35.
Therefore, Dealership A sold 35 trucks.
a) The equation for the total cars and trucks sold by Dealership A is:
Total cars and trucks = 164
b) The equation for the total cars and trucks sold by Dealership B can be derived using the given information that Dealership B sells twice as many cars and half as many trucks as Dealership A. Let "x" represent the number of cars sold by Dealership A and "y" represent the number of trucks sold by Dealership A. Therefore, the equation for Dealership B would be:
Total cars and trucks = 2x + (1/2)y = 229
c) To find the number of cars sold by Dealership A, we use the equation from part a:
Total cars and trucks = 164
Since this equation represents the sum of cars and trucks sold by Dealership A, we can substitute the total number of cars in place of the sum. Therefore:
164 = Number of cars sold by Dealership A
Therefore, Dealership A sold 164 cars.
d) Considering that Dealership B sold half as many trucks as Dealership A, we use the equation from part b to determine the number of trucks sold by Dealership B. From the equation:
2x + (1/2)y = 229
We want to find the value of "y" (number of trucks sold by Dealership B). Since Dealership B sold half as many trucks as Dealership A, we substitute 164 (number of trucks sold by Dealership A) divided by 2 into the equation:
2x + (1/2) * (164/2) = 229
Simplifying the equation:
2x + 82 = 229
Subtracting 82 from both sides:
2x = 147
Dividing both sides by 2:
x = 73.5
Therefore, Dealership B sold 73.5 trucks.
To solve this problem, we will start by setting up two variables: x for the number of cars sold at Dealership A and y for the number of trucks sold at Dealership A.
a) The equation for the total cars and trucks at Dealership A can be written as:
x + y = 164
b) Now, let's set up the equation for Dealership B. We know that it sells twice as many cars as Dealership A and half as many trucks. Therefore, the equation can be written as:
(2x) + (1/2)y = 229
c) To find the number of cars sold at Dealership A, we can solve the equation x + y = 164. This can be done manually or by using the Desmos Graphing Calculator.
1. Go to the Desmos Graphing Calculator website (https://www.desmos.com/calculator).
2. In the first equation box, type "x + y = 164".
3. Press Enter or click on the "+" button to plot the graph.
4. Observe the intersection point of the graph with the x-axis. The x-coordinate of this point represents the number of cars sold at Dealership A. In this case, the x-coordinate will be the answer to the question.
d) To find the number of trucks sold at Dealership B, we need to consider that Dealership B sold half as many trucks as Dealership A. Therefore, the number of trucks sold at Dealership B will be (1/2)y. Using this information, we can solve the equation (2x) + (1/2)y = 229.
1. In the second equation box on the Desmos Graphing Calculator, type "(2x) + (1/2)y = 229".
2. Press Enter or click on the "+" button to plot the graph.
3. Observe the intersection point of the graph with the y-axis. The y-coordinate of this point represents the number of trucks sold at Dealership B. In this case, the y-coordinate will be the answer to the question.
Using the graphing calculator helps visualize the equations and provides a quick way to find the solutions.