`-x + y = -22, 3x + 4y = 4, 4x - 8y = 32` Use matricies to solve the system of equations. Use Gaussian elimination with back-substitution.

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The augmented matrix is `[[-1,1,-22],[3,4,4],[4,-8,32]]`

On applying `R_1 -gt -R_1 ` and `R_3 -gt (R_3)/4` we get (means inverse the sign of first row and divide the third row by 4)

`[[1,-1,22],[3,4,4],[1,-2,8]]`

On applying `R_2 -gt 3R_1 - R_2` and `R_3 -gt R_3 - R_1` we get

`[[1,-1,22],[0,-7,62],[0,-1,-14]]`

On applying...

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The augmented matrix is `[[-1,1,-22],[3,4,4],[4,-8,32]]`

On applying `R_1 -gt -R_1 ` and `R_3 -gt (R_3)/4` we get (means inverse the sign of first row and divide the third row by 4)

`[[1,-1,22],[3,4,4],[1,-2,8]]`

On applying `R_2 -gt 3R_1 - R_2` and `R_3 -gt R_3 - R_1` we get

`[[1,-1,22],[0,-7,62],[0,-1,-14]]`

On applying `R_3 -gt 7R_3 - R_2` we get 

`[[1,-1,22],[0,-7,62],[0,0,-160]]`

On applying `R_2 -gt (R_2)/(-7) ` we get

`[[1,-1,22],[0,1,-62/7],[0,0,-160]]`

Hence the given system of equations is equivalent to the following

`x - y =22`

`y = -62/7 ` and `0(x) + 0(y) =-160`

Clearly no `x, y` satisfy the last equation. 

Hence the given system has no solution and is inconsistent

 

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