A company owns two dealerships, both of which sell cars and trucks. The first dealership sells a total of 164 cars and trucks. The second dealership sells twice as many cars and half as many trucks as the first dealership, and sells a total of 229 cars and trucks. The system of equations below represents this situation.
How many cars does the first dealership sell?
first dealer sells c cars and t trucks
c + t = 164
second dealer
2 c + (1/2) t = 229 or 4 c + t = 458
so
4 c + t = 458
c + t = 164
-----------------------subtract
3 c = 294
c = 98 cars by first dealer
To find out how many cars the first dealership sells, we can set up a system of equations based on the given information. Let's denote the number of cars sold by the first dealership as "C1" and the number of trucks sold as "T1". Similarly, let's denote the number of cars sold by the second dealership as "C2" and the number of trucks sold as "T2".
From the information given, we can set up the following equations:
Equation 1: C1 + T1 = 164 (The total number of cars and trucks sold by the first dealership is 164)
Equation 2: C2 + T2 = 229 (The total number of cars and trucks sold by the second dealership is 229)
We also know that the second dealership sells twice as many cars and half as many trucks as the first dealership. This can be translated into the following equations:
Equation 3: C2 = 2C1 (The number of cars sold by the second dealership is twice the number of cars sold by the first dealership)
Equation 4: T2 = 1/2T1 (The number of trucks sold by the second dealership is half the number of trucks sold by the first dealership)
Now we can solve this system of equations to find the values of C1, T1, C2, and T2.
From equation 4, we can substitute T2 with 1/2T1 in equation 2:
C2 + 1/2T1 = 229
From equation 3, we can substitute C2 with 2C1 in this new equation:
2C1 + 1/2T1 = 229
Rearranging the terms:
5/2C1 + 1/2T1 = 229
Now, let's substitute C1 + T1 = 164 (from equation 1) in the above equation:
5/2C1 + 1/2(164 - C1) = 229
Simplifying:
5/2C1 + 82 - 1/2C1 = 229
Combining like terms:
3/2C1 + 82 = 229
Subtracting 82 from both sides:
3/2C1 = 147
Multiplying both sides by 2/3:
C1 = 98
Therefore, the first dealership sells 98 cars.
Let's assign variables to represent the number of cars and trucks sold at the first dealership.
Let c represent the number of cars sold at the first dealership.
Let t represent the number of trucks sold at the first dealership.
According to the given information, we can write the following equations:
Equation 1: c + t = 164 (the total number of cars and trucks sold at the first dealership is 164)
Now let's form an equation using the information given about the second dealership:
The second dealership sells twice as many cars as the first dealership, so the number of cars sold at the second dealership is 2c.
The second dealership sells half as many trucks as the first dealership, so the number of trucks sold at the second dealership is t/2.
Equation 2: 2c + (t/2) = 229 (the total number of cars and trucks sold at the second dealership is 229)
Now we have a system of two equations:
Equation 1: c + t = 164
Equation 2: 2c + (t/2) = 229
To solve this system of equations, we can use substitution or elimination methods. Let's use the substitution method.
From Equation 1, we can isolate c:
c = 164 - t
Now substitute this expression for c into Equation 2:
2(164 - t) + (t/2) = 229
Multiply through by 2 to remove the fraction:
328 - 2t + t/2 = 229
Multiply through by 2 to get rid of the fraction:
656 - 4t + t = 458
Combine like terms:
4t - t = 656 - 458
3t = 198
Divide by 3:
t = 198/3
t = 66
Now substitute the value of t back into Equation 1 to find c:
c + 66 = 164
c = 164 - 66
c = 98
Therefore, the first dealership sells 98 cars.