Solve 103x + 86y + 79z = 1040.25            48x + 32y + 26z =  406.50            45x + 25y + 18z =  334.00 use equation for gaussian or gauss-jordan elimination to get price of...

Solve 103x + 86y + 79z = 1040.25

           48x + 32y + 26z =  406.50

           45x + 25y + 18z =  334.00

use equation for gaussian or gauss-jordan elimination to get price of each drink sold above

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Write the three equations as an augmented matrix:

103  86  79  |  1040.25

48   32   26  |  406.5

45   25   18  |  334


Gaussian elimination should result in an augmented matrix of the (echelon) form:

1 a b | c      or  1 a b | c

0 1 d | e          0 0 1 | d

0 0 1 | f           0 0 0 | e


First divide row 1 by 103:

1    0.835   0.767    |  10.100

48  32         26        |  406.5

45  25         18        |  334

Then take 48*row 1 from row 2 and 45*row 1 from row 3:

1   0.835    0.767   | 10.100

0  -0.808   -10.816 | -78.277

0  -12.573 -16.515 | -120.478

Now multiply row 2 by -1/0.808:

1   0.835     0.767    | 10.100

0   1           1.339    | 9.691

0  -12.573  -16.515  | -120.478

Now add 12.573*row 2 to row 3:

1  0.835   0.767   | 10.100

0  1         1.339   | 9.691

0  0         0.320   | 1.359

Finally multiply row 3 by 1/0.320

1  0.835   0.767  | 10.100

0  1         1.339  | 9.691

0  0         1        | 4.250

Gauss-jordan elimination results in an augmented matrix with the identity matrix on the lefthand side.

Take 1.339*row 3 from row 2 and 0.767*row 3 from row 1:

1  0.835   0   | 6.840

0  1         0   | 4.000

0  0          1  | 4.250

Finally take 0.835*row 2 from row 1:

1  0  0  | 3.500

0  1  0  | 4.000

0  0  1  | 4.250

`therefore`  ` x = $3.50`, `y = $4.00`  and `z = $4.25` answer

 

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