Answer: 6 apple pies and 4 shoo-fly pies were made.
Explanation:
Let's assume the number of apple pies made is A, and the number of shoo-fly pies made is S.
We know that there were 8 slices in each apple pie, so the total number of apple pie slices would be 8A. Similarly, the total number of shoo-fly pie slices would be 10S.
We also know that the total number of slices available was 84. So we can write an equation:
8A + 10S = 84
We have two unknowns and only one equation, so we need another equation to solve for A and S. We know that the siblings made a total of 10 pies. So we can write another equation:
A + S = 10
Now we have two equations and two unknowns, so we can solve for A and S. We can use substitution to eliminate one variable:
A + (10 - A) = 10
10 = 2A + 10
2A = 0
A = 0
This is obviously not the solution we're looking for, so there must be an error in our calculations. Let's check our first equation:
8A + 10S = 84
If A = 0, then we have:
10S = 84
S = 8.4
This is also not a valid solution since we can't make 8.4 shoo-fly pies. The mistake we made was assuming that both A and S were whole numbers. We can fix this by using another equation:
A + S = 10
We know that A and S are both whole numbers and that A + S = 10. The only pairs of whole numbers that add up to 10 are (1, 9), (2, 8), (3, 7), (4, 6), and (5, 5).
Let's try each pair and see which one gives us a valid solution:
(1, 9): 8(1) + 10(9) = 98 (not 84)
(2, 8): 8(2) + 10(8) = 96 (not 84)
(3, 7): 8(3) + 10(7) = 94 (not 84)
(4, 6): 8(4) + 10(6) = 92 (not 84)
(5, 5): 8(5) + 10(5) = 90 (not 84)
None of the pairs work, which means there is no valid solution that uses whole numbers for A and S.
However, we can use decimals to get a solution that's close to the desired number of slices. Let's try (4.2, 5.8):
8(4.2) + 10(5.8) = 84.4 (close to 84)
This means that the siblings made 4.2 apple pies and 5.8 shoo-fly pies. Since we can't make a fraction of a pie, we'll round up the number of apple pies and round down the number of shoo-fly pies:
4 apple pies and 5 shoo-fly pies would give us a total of 8(4) + 10(5) = 84 slices, which is the desired number.