Use gaussian elimination to solve the system of equations: x1+x2-x3-2x4=-4 , 2x1-x2+3x3+3x4=5 , x2+7x3+3x4=2 , 2x1+4x2+2x3+x4=3                               

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The set of linear equations

x1+x2-x3-2x4=-4
2x1-x2+3x3+3x4=5
x2+7x3+3x4=2
2x1+4x2+2x3+x4=3

has to be solved.

Create the augmented matrix form

 

`[[1,1,-1,-2|-4],[2,-1,3,3|5],[0,1,7,3|2],[2,4,2,1|3]] `

=> `[[1,0,-8,-5|-6],[2,-1,3,3|5],[0,1,7,3|2],[2,4,2,1|3]]`

=> `[[1,0,-8,-5|-6],[0,-1,19,13|17],[0,1,7,3|2],[2,4,2,1|3]]`

=> `[[1,0,-8,-5|-6],[0,1,-19,-13|-17],[0,1,7,3|2],[0,4,18,11|15]]`

=> `[[1,0,-8,-5|-6],[0,1,-19,-13|-17],[0,0,26,16|19],[0,4,18,11|15]]`

=> `[[1,0,-8,-5|-6],[0,1,-19,-13|-17],[0,0,1,8/13|19/26],[0,4,18,11|15]]`

=> `[[1,0,-8,-5|-6],[0,1,-19,-13|-17],[0,0,1,8/13|19/26],[0,0,94,63|83]]`

=> `[[1,0,-8,-5|-6],[0,1,-19,-13|-17],[0,0,1,8/13|19/26],[0,0,0,67/13|186/13]]`

=> `[[1,0,-8,-5|-6],[0,1,-19,-13|-17],[0,0,1,8/13|19/26],[0,0,0,1|186/67]]`

=> `[[1,0,0,-1/13|-2/13],[0,1,-19,-13|-17],[0,0,1,8/13|19/26],[0,0,0,1|186/67]]`

=> `[[1,0,0,0|4/67],[0,1,-19,-13|-17],[0,0,1,8/13|19/26],[0,0,0,1|186/67]]`

=> `[[1,0,0,0|4/67],[0,1,0,-17/13|-81/26],[0,0,1,8/13|19/26],[0,0,0,1|186/67]]`

=> `[[1,0,0,0|4/67],[0,1,0,0|69/134],[0,0,1,8/13|19/26],[0,0,0,1|186/67]]`

=> `[[1,0,0,0|4/67],[0,1,0,0|69/134],[0,0,1,0|-131/134],[0,0,0,1|186/67]]`

This gives the solution of the set of equations as x1 = 4/67, x2 = 69/134, x3 = -131/134 and x4 = 186/67.

Approved by eNotes Editorial Team