Question 1181659: Ron attends a cocktail party (with his graphing calculator in his pocket). He wants to limit his food intake to 117 g protein, 100 g fat, and 150 g carbohydrate. According to the health conscious hostess, the marinated mushroom caps have 3 g protein, 5 g fat, and 9 g carbohydrate; the spicy meatballs have 14 g protein, 7 g fat, and 15 g carbohydrate; and the deviled eggs have 13 g protein, 15 g fat, and 6 g carbohydrate. How many of each snack can he eat to obtain his goal?
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13203)   (Show Source):
You can put this solution on YOUR website!
Let x = # of mushroom caps
let y = # of spicy meatballs
let z = # of deviled eggs
The equations are
3x+14y+13z = 117 (protein)
5x+7y+15z = 100 (fat)
9x+15y+6z = 150 (carbs)
There are dozens of ways of solving systems of equations like this....
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.
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.
So y = 5.
Then the three equations are
3x+13z=47 (protein)
5x+15z=65 (fat)
9x+6z=75 (carbs)
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.
10x+30z=130
45z+30z=375
------------
35x=245
x = 245/35 = 7
10(7)+30z=130
70+30z=130
30z=60
z=2
ANSWER:
x=7 mushroom caps
y=5 spicy meatballs
z=2 deviled eggs
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
Answer by ikleyn(52812)  (Show Source):
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