Algebra II matrices word problem

Question

Math 171 matrix problems

 

A hospital dietitian must plan a dinner menu that provides 35g of protein, 50.5g of carbohydrates, and 370mg of calcium. The hospital food service tells the dietitian that the dinner today will be chicken, corn, and milk. Each serving of chicken has 40g of protein, 15g of carbohydrates, and 100mg of calcium. Each servingof corn has 3g of protein, 16g of carbohydrates, and 10mg of calcium. Each glass of milk has 9g of protein, 11g of carbohydrates, and 300mg of calcium. How many servings of each are required to provide the desired nutritional values?

Answer 1

x is the chicken

y is the corn

z is the milk

40x + 3y + 9z = 35

15x + 16y + 11z = 50.5

100x + 10y + 300z = 370

The solutions are x=1/2, y = 2, z = 1

1/2 serving of chicken, 2 serving of corn, one serving of milk

First, multiplies second equation by 2 to clear the decimal:

40x + 3y + 9z = 35

30x + 32y + 22z = 101

100x + 10y + 300z = 370

Next multiplies the first equation by -3/4 and adds to the second equation:

40x + 3y + 9z = 35

119/4 y + 61/4z = 299/4 <-- 32*4/4 - 9/4 = 119/4 and (88-27)/4=61/4 and (404-105)/4 = 299/4

100x + 10y + 300z = 370

Then multiplies the first equation by -2.5 and adds to the third equation:

40x + 3y + 9z = 35

119/4 y + 61/4z = 299/4

2.5y +277.5z = 282.5

Multiplies the 2nd equation by 4 and the third equation by 2:

40x + 3y + 9z = 35

119y + 61z = 299

5y + 555z = 565

Solves equations 2 and 3 by elimination:

-5( 119y + 61z = 299)

119 ( 5y + 555z = 565)

-595 y + -305z = -1495

595y + 66045 z = 67235

65740 z = 65740

z = 1

The third equation in bold above forces y=2

The original first equation forces x=1/2