Question 1181659
<br>
Let x = # of mushroom caps
let y = # of spicy meatballs
let z = # of deviled eggs<br>
The equations are<br>
3x+14y+13z = 117  (protein)
5x+7y+15z = 100  (fat)
9x+15y+6z = 150 (carbs)<br>
There are dozens of ways of solving systems of equations like this....<br>
I would use the fact that x, y, and z are non-negative integers to see if there is a quick path to the solution.<br>
And indeed I see in the equation for fat that "5x", "15z", and "100" are all multiples of 5 -- and that means "7y" must be a multiple of 5.  0 is not a likely value for y; and y=10 in both the equation for protein and the equation for carbs doesn't make sense.<br>
So y = 5.<br>
Then the three equations are<br>
3x+13z=47  (protein)
5x+15z=65 (fat)
9x+6z=75 (carbs)<br>
A quick look at the coefficients in those three equations shows that probably the fastest path to the final answer is to eliminate z between the equations for fat and carbs.<br>
10x+30z=130
45z+30z=375
------------
35x=245
x = 245/35 = 7<br>
10(7)+30z=130
70+30z=130
30z=60
z=2<br>
ANSWER:
x=7 mushroom caps
y=5 spicy meatballs
z=2 deviled eggs<br>
CHECK:
protein: 3(7)+14(5)+13(2)=21+70+26 = 117
fat: 5(7)+7(5)+15(2)=35+35+30=100
carbs: 9(7)+15(5)+6(2)=63+75+12=150<br>