Question

How do you solve these set of linear equations: `3x + 2y + z - 7; 5x + 5y + 4z = 3; 3x + 2y + 3z = 1`?

Answer 1

Write an augmented matrix, perform row operations, until you obtain an identity matrix, read the x, y, and z values from the column vector:
`x = 4, y = -1 and z = -3`

Explanation:

Please allow me to apologise in advance. The formatting does not seem to do well with matrices. They do not look good without formatting but the formatting makes it look much worse. If anyone knows how to do matrix good looking formatting, please send me a note.

Let's begin by making an augmented matrix. This a square matrix of the coefficients x, y, and z with a column vector of the values on the right of the equal sign.

Write the row for `5x + 5y + 4z = 3`

|5 5 4||3|

Add the row for `3x + 2y + 3z = 1`

|5 5 4||3|
|3 2 3||1|

Add the row for `3x + 2y + z = 7`

|5 5 4||3|
|3 2 3||1|
|3 2 1||7|

Subtract row 2 from row 1:

|2 3 1||2|
|3 2 3||1|
|3 2 1||7|

Subtract row 3 from row 1:

|-1 1 0||-5|
|3 2 3||1|
|3 2 1||7|

Multiply row 1 by 3 and add to row 2:

|-1 1 0||-5|
|0 5 3||-14|
|3 2 1||7|

Multiply row 1 by 3 and add to row 3:

|-1 1 0||-5|
|0 5 3||-14|
|0 5 1||-8|

Subtract row 2 from row 3:

|-1 1 0||-5|
|0 5 3||-14|
|0 0 -2||6|

Divide row 3 by -2:

|-1 1 0||-5|
|0 5 3||-14|
|0 0 1||-3|

Multiply row 3 by -3 and add to row 2:

|-1 1 0||-5|
|0 5 0||-5|
|0 0 1||-3|

Divide row 2 by 5:

|-1 1 0||-5|
|0 1 0||-1|
|0 0 1||-3|

Multiply row 1 by -1:

|1 -1 0||5|
|0 1 0||-1|
|0 0 1||-3|

Add row 2 to row 1:

|1 0 0||4|
|0 1 0||-1|
|0 0 1||-3|

We have obtained an identity matrix (a matrix with 1s on the main diagonal and zeros, elsewhere). We read the values for x, y, and z from the column vector on the right: `x = 4, y = -1 and z = -3`