Question

How do you solve using gaussian elimination or gauss-jordan elimination, `-x + y +2z = 1`, `2x -2z = 0`, `2x + y + 2z = 0`?

Answer 1

`(x,y,z)=(-1/3,1 1/3, -1/3)`
`color(white)("XXXXXXXX")`see below for solution using Gauss-Jordan method

Explanation:

Convert the given equations into augmented matrix form;
then work a diagonal of "pivot entries" from the top left reducing the left side of the matrix to an identity sub-matrix (Gauss-Jordan form).

Initial Augmented Matrix:
#( (color(white)("XXXX")-1,color(white)("XXXXX")1,color(white)("XXXXX")2,color(white)("XXXXX")1), (color(white)("XXXXX")2,color(white)("XXXXX")0,color(white)("XXXX")-2,color(white)("XXXXX")0), (color(white)("XXXXX")2,color(white)("XXXXX")1,color(white)("XXXXX")2,color(white)("XXXXX")0) )#

Pivot Row: 1 Pivot Column: 1
pivot Row reduced:
#( (color(white)("XXXXX")1,color(white)("XXXX")-1,color(white)("XXXX")-2,color(white)("XXXX")-1), (color(white)("XXXXX")2,color(white)("XXXXX")0,color(white)("XXXX")-2,color(white)("XXXXX")0), (color(white)("XXXXX")2,color(white)("XXXXX")1,color(white)("XXXXX")2,color(white)("XXXXX")0) )#

Non-pivot rows reduced for pivot column
#( (color(white)("XXXXX")1,color(white)("XXXX")-1,color(white)("XXXX")-2,color(white)("XXXX")-1), (color(white)("XXXXX")0,color(white)("XXXXX")2,color(white)("XXXXX")2,color(white)("XXXXX")2), (color(white)("XXXXX")0,color(white)("XXXXX")3,color(white)("XXXXX")6,color(white)("XXXXX")2) )#

Pivot Row: 2 Pivot Column: 2
pivot Row reduced:
#( (color(white)("XXXXX")1,color(white)("XXXX")-1,color(white)("XXXX")-2,color(white)("XXXX")-1), (color(white)("XXXXX")0,color(white)("XXXXX")1,color(white)("XXXXX")1,color(white)("XXXXX")1), (color(white)("XXXXX")0,color(white)("XXXXX")3,color(white)("XXXXX")6,color(white)("XXXXX")2) )#

Non-pivot rows reduced for pivot column
#( (color(white)("XXXXX")1,color(white)("XXXXX")0,color(white)("XXXX")-1,color(white)("XXXXX")0), (color(white)("XXXXX")0,color(white)("XXXXX")1,color(white)("XXXXX")1,color(white)("XXXXX")1), (color(white)("XXXXX")0,color(white)("XXXXX")0,color(white)("XXXXX")3,color(white)("XXXX")-1) )#

Pivot Row: 3 Pivot Column: 3
pivot Row reduced:
#( (color(white)("XXXXX")1,color(white)("XXXXX")0,color(white)("XXXX")-1,color(white)("XXXXX")0), (color(white)("XXXXX")0,color(white)("XXXXX")1,color(white)("XXXXX")1,color(white)("XXXXX")1), (color(white)("XXXXX")0,color(white)("XXXXX")0,color(white)("XXXXX")1,color(white)("XXXX")-0.333) )#

Non-pivot rows reduced for pivot column
#( (color(white)("XXXXX")1,color(white)("XXXXX")0,color(white)("XXXXX")0,color(white)("XXXX")-0.333), (color(white)("XXXXX")0,color(white)("XXXXX")1,color(white)("XXXXX")0,color(white)("XXXXX")1.333), (color(white)("XXXXX")0,color(white)("XXXXX")0,color(white)("XXXXX")1,color(white)("XXXX")-0.333) )#