Solution to the Problem:
There are two answers:
Answer #1
1 quail egg at 5 quadrons each for a total of 5 quadrons
22 hen eggs at 4 quadrons each for a total of 88 quadrons
1 goose egg at 10 quadrons each for a total of 10 quadrons
Answer #2
7 quail eggs at 3 quadrons each for a total of 21 quadrons
10 hen eggs at 4 quadrons each for a total of 40 quadrons
7 goose eggs at 6 quadrons each for a total of 42 quadrons
I used three variables to solve the problem, but I noticed that there were only two equations, so that alerted me that there may be more than one answer.
Let x = # of quail eggs
Let h = # of hen eggs
Then x also equals the # of goose eggs
Let y = cost of quail eggs
We are told that 4 = cost of hen eggs
Then 2y = cost of the goose eggs.
My two equations are:
x + h + x = 24 and
xy + 4h + 2xy = 103
From equation 1, we get h = 24 - 2x
This tells me that x must be between 0 and 12 in order for h to be a non-negative number.
From the second equation, I get the following:
3xy + 4h = 103
Substituting equation 1 in equation 2, we get
3xy + 4(24 - 2x) = 103
3xy + 96 - 8x = 103
Solving for y:
y = (8x + 7) / 3x
Now, I set up a table for these three variables for the values in the domain (using the equations above):
| x
|
h
|
y
|
works?
|
| 0
|
24
|
7
|
no
|
| 1
|
22
|
5
|
yes
|
| 2
|
20
|
3.83
|
no
|
| 3
|
18
|
3.44
|
no
|
| 4
|
16
|
3.25
|
no
|
| 5
|
14
|
3.13
|
no
|
| 6
|
12
|
3.05
|
no
|
| 7
|
10
|
3
|
yes
|
| 8
|
8
|
2.95
|
no
|
| 9
|
6
|
2.92
|
no
|
| 10
|
4
|
2.9
|
no
|
| 11
|
2
|
2.87
|
no
|
| 12
|
0
|
2.86
|
no
|
So, the answers are:
#1
1 quail egg at 5 quadrons each for a total of 5 quadrons
22 hen eggs at 4 quadrons each for a total of 88 quadrons
1 goose egg at 10 quadrons each for a total of 10 quadrons
24 eggs for 103 quadrons
#2
7 quail eggs at 3 quadrons each for a total of 21 quadrons
10 hen eggs at 4 quadrons each for a total of 40 quadrons
7 goose eggs at 6 quadrons each for a total of 42 quadrons
24 eggs for 103 quadrons