Set up and solve a system of equations for each problem
John and Belinda are selling pies for a school fundraiser. Customers can buy apple pies and cherry pies.
John sold 3 apple pies and 9 cherry pies for a total of $183.
Belinda sold 6 apple pies and 13 cherry pies for a total of $276. What is the cost each of one apple pies and one cherry pies?
3a + 9c = 183
... 6a + 18c = 366
6a + 13c = 276
subtract equations to eliminate a
solve for c, then substitute back to find a
To set up and solve a system of equations for this problem, we need to assign variables to the unknown quantities. Let's say the cost of one apple pie is 'a' dollars and the cost of one cherry pie is 'c' dollars.
Based on the given information, we can set up two equations:
1) John sold 3 apple pies and 9 cherry pies for a total of $183:
3a + 9c = 183
2) Belinda sold 6 apple pies and 13 cherry pies for a total of $276:
6a + 13c = 276
Now we have a system of two equations with two variables. We can solve this system to find the values of 'a' and 'c'.
We can use either the substitution or elimination method to solve this system. Let's use the elimination method:
Multiply the first equation by 2, and multiply the second equation by 3 to eliminate the 'a' terms:
6a + 18c = 366
18a + 39c = 828
Now, subtract the first equation from the second equation:
(18a + 39c) - (6a + 18c) = 828 - 366
This simplifies to:
12a + 21c = 462
Now we have a new equation:
12a + 21c = 462
To make the coefficients of 'a' and 'c' easier to work with, divide the equation by 3:
4a + 7c = 154
We now have two simplified equations:
4a + 7c = 154 (Equation 1)
3a + 9c = 183 (Equation 2)
From Equation 2, we can isolate 'a' by multiplying the entire equation by 4/3:
(4/3)(3a + 9c) = (4/3)(183)
4a + 12c = 244
Now we can subtract Equation 1 from this new equation:
(4a + 12c) - (4a + 7c) = 244 - 154
12c - 7c = 90
This simplifies to:
5c = 90
Divide both sides of the equation by 5:
c = 18
Now that we have the value of 'c' (the cost of one cherry pie), we can substitute it back into Equation 1 to find the value of 'a' (the cost of one apple pie):
4a + 7(18) = 154
4a + 126 = 154
Subtract 126 from both sides:
4a = 28
Finally, divide both sides of the equation by 4:
a = 7
Therefore, the cost of one apple pie is $7, and the cost of one cherry pie is $18.