system equation

Question

Now let us consider fat grams in pepperoni pizza slices from three popular pizza restaurants. The number of fat grams in a Pizza Hut slice exceeds the fat grams in a Domino's slice by 0.5g. If you add together the fat grams in a Pizza Hut slice and a Domino's slice, it is 1.5g more than twice the number of fat grams in a Little Caesar's slice. Three times the amount of the fat grams in all three is 49.5g. Find the fat content of each slice of pizza using the matrix method.

Answer 1

P = D + 0.5 => P - D = 0.5

 

P + D = 2C + 1.5 => P + D - 2C = 1.5

 

3P + 3D + 3C = 49.5

 

Using Augmented Matrix and Row Operations:

 

R1 = {1,-1,0,0.5}

R2 = {1,1,-2,1.5}

R3 = {3,3,3,49.5}

 

R1 >> 2 R1

R2 >> 2 R2

R3  >> 2 R3

 

R1 = {2,-2,0,1}
R2 = {2,2,-4,3}
R3 = {6,6,6,99}

 

R2 >> R2 - R1

R3 >> R3 - 3 R1

 

R1 = {2,-2,0,1}
R2 = {0,4,-4,2}
R3 = {0,12,6,96}

 

R3 >> R3 - 3 R2

 

R1 = {2,-2,0,1}
R2 = {0,4,-4,2}
R3 = {0,0,18,90}

 

R2 >> R2/2

R3 >> R3/18

 

R1 = {2,-2,0,1}
R2 = {0,2,-2,1}
R3 = {0,0,1,5}

 

R2 >> R2 + 2 R3

 

R1 = {2,-2,0,1}
R2 = {0,2,0,11}
R3 = {0,0,1,5}

 

R1 >> R1 + R2

 

R1 = {2,0,0,12}
R2 = {0,2,0,11}
R3 = {0,0,1,5}

 

R1 >> R1/2

R2 >> R2/2

 

R1 = {1,0,0,6}
R2 = {0,1,0,5.5}
R3 = {0,0,1,5}

 

{6,5.5,5} grams of fat in piece of pizza at {Pizza Hut,Domino's,Little Caesar's}.

 

Checked with GeoGebra:

http://www.wyzant.com/resources/files/261883/augmented_matrix_row_ops

 

The augmented matrix in the GeoGebra solution included an additional 3 columns containing an identity matrix.

 

So it was [ A | C | I ],

 

And the result of the reduction to row eschelon form was

[ I | S | A^(-1) ].

 

So you could solve the system with this matrix equation:

 

S = A^(-1) C