Rabbits in a lab are to be kept on a strict daily diet that includes 47 grams of protein, 26 grams of fat and 44 grams of carbs.

Question

The scientist has only three FoodMaxx is available with the following grams of nutrients per unit:

Mix A: 6 protein, 4 fat, 2 carbs

Mix B: 4 protein, 2 fat, 6 carbs

Mix C: 5 protein, 2 fat, 8 carbs

Fine how many units of each mix are needed daily to meet each rabbits dietary needs.

Answer 1

6a + 4b + 5c = 47

4a + 2b + 2c = 26

2a + 6b + 8c = 44

divides 2nd and 3rd equations by 2:

6a + 4b + 5c = 47

2a + b + c = 13

a + 3b + 4c = 22

Eliminates A:

Pairs equations #1 and #2:

6a + 4b + 5c = 47

2a + b + c = 13

-3 * Equation #2 + Equation #1:

b + 2c = 8

Pairs equations #2 and #3:

2a + b + c = 13

a + 3b + 4c = 22

-2 * equation #3 + equation #2:

-5b + -7c = -31

5b + 7c = 31 <--- multiplies everything by -1

2 by 2 system from equations above in bold:

b + 2c = 8

5b + 7c = 31

solves by substitution: b = 8 - 2c

5(8 - 2c) + 7c = 31

40 - 10c + 7c = 31

40 - 3c = 31

-3c = -9

c = 3

Then b = 8 - 2(3) = 8 - 6 = 2

Simplified second equation: 2a + b + c = 13

2a + 2 + 3 = 13

2a = 8

a = 4

Check:

6(4) + 2*4 + 3*5 = 24 + 8 + 15 = 47

4*4 + 2*2+ 2*3 = 16 + 4 + 6 = 26

4*2 + 6*2 + 3*8 = 8 + 12+ 24 = 44

A=4 B=2 C=3