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

Jul 26, 2016

$\left(x , y , z\right) = \left(- \frac{1}{3} , 1 \frac{1}{3} , - \frac{1}{3}\right)$
$\textcolor{w h i t e}{\text{XXXXXXXX}}$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) )