`3x + 2y - z + w = 0, x - y + 4z + 2w = 25, `

`-2x + y + 2z - w = 2, x + y + z + w =6`

Use matricies to solve the system of equations. Use Gaussian elimination with back-substitution.

Expert Answers

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The above system of equations can be represented by the coefficient matrix A and right hand side vector b as follows:



The augmented matrix can be written as:

`[A b]=[[3,2,-1,1,0],[1,-1,4,2,25],[-2,1,2,-1,2],[1,1,1,1,6]]`

Now let's bring the above matrix in row-echelon form by performing various row operations,

Rewrite the 1st Row`(R_1)` as `(R_1+R_3)`  


Rewrite the 2nd Row`(R_2)` as `(R_2-R_4)`


Rewrite the 3rd Row`(R_3)` as`(R_3+2R_4)`


Rewrite the 4th Row as `(R_4-R_1)`


Rewrite the 2nd Row as `(R_2+R_3)`


Rewrite the 3rd Row as `(R_3-3R_2)`


Rewrite the 4th Row as `(R_4+2R_2)`


Rewrite the 3rd Row as `(R_3+R_4)`


Rewrite the 3rd Row by dividing it with -3,


Rewrite the 4th Row as`(R_4-14R_3)`


Rewrite the 4th Row by dividing it with 5,


Now the above matrix is row-echelon form and we can perform back substitution on the corresponding system,

`x+3y+z=2`     ----- Eq:1

`y+7z+2w=33`    ------ Eq:2



Substitute back the value of z and w in Eq:2,




Substitute back the value of y and z in Eq:1,





So the solutions are x=3,y=-2,z=5 and w=0


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