Solve by matrix method

Question

Food I has 3 units of vitamins A, 9 units of vitamin B and 12 units of vitamin C. Food II has 6 units of vitamins A, 9 units of vitamin B and 15 units of vitamin C. Food III has 9 units of vitamins A, 0 units of vitamin B and 9 units of vitamin C. The number of units of vitamins A,B and C required are 33,27 and 60 respectively. Find the amount of three foods that will provide exactly these amounts. Use matrix method

Answer 1

x is the amount of food1

y is the amount of food2

z is the amount of food3

3x + 6y + 9z = 33

9x + 9y + 0z = 27

12x + 15y + 9z = 60

3 6 9 33

9 9 0 27

12 15 9 60

divides everything by 3

1 2 3 11

1 1 0 3

4 5 3 20

row1 = -row2 + row1;

row3 = -4*row2 + row3;

0 1 3 8

1 1 0 3

0 1 3 8

row3 = -row1 + row3

and then there are infinitely many solutions

Per 2nd equation: x+y = 3

y = 3-x

Substitutes into first and third equations:

x + 2(3-x) + 3z = 11

x + 6 - 2x + 3z = 11

-x + 3z + 6 = 11

-x + 3z = 5

4x + 5(3-x) + 3z = 20

4x + 15 - 5x + 3z = 20

-x + 3z = 5

So for free variable x

y = 3-x

z = (5+x)/3

For example, if x=0 then only food 2 and 3 are used

which results in y = 3-0 = 3 servings of food2 and z=(5+0)/3 = 5/3

servings of food 3

Vitamin A = 6*3 + (5/3)*9 = 18 + 15 = 33

Vitamin B = 9*3 + (5/3)*0 = 27

Vitamin C = 15*3 + (5/3)*9 = 45 + 15 = 60

which is the presribed amount

Now for x=1, then y = 3-1 = 2 servings of food2 and z = (5+1)/3 = 2

servings for food 3

Vitamin A = 3*1 + 6*2 + 9*2 = 3 + 12 + 18 = 33

Vitamin B = 9*1 + 9*2 + 0 * 9 = 9 + 18 = 27

Vitamin C = 12*1 + 15*2 + 9*2 = 12 + 30 + 18 = 60

So there are 2 solutions and any number of solutions

can be created by picking a value of x, calculating

y and z in this manner