`-x + y - z = -14, 2x - y + z = 21, 3x + 2y + z = 19` Use matricies to solve the system of equations (if possible). Use Gauss-Jordan elimination.

The augmented matrix is

`[[-1,1,-1,-14], [2,-1,1,21],[3,2,1,19]] `

On applying `R_1 -gt R_1 +R_2` we get (means changing 1st row as the sum of  of first and second row)

`[[1,0,0,7],[2,-1,1,21],[3,2,1,19]] `

On applying `R_2 -gt R_2 - 2R_1` we get

`[[1,0,0,7], [0,-1,1,7], [3,2,1,19]] `

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

`[[1,0,0,7],[0,1,-1,-7],[0,2,1,-2]] `

On applying `R_3 -gt R_3 - 2R_2` we get

`[[1,0,0,7],[0,1,-1,7],[0,0,3,12]] `

On applying `R_3 -gt(R_3)/3` we get

`[[1,0,0,7],[0,1,-1,-7],[0,0,1,4]] `

Hence the given system of equations is equivalent to the following system of equations

`x = 7 `

`y - z = -7 ` and

`z = 4 `

`therefore` the solution set is

`x = 7, y =-3, z = 4 `

Approved by eNotes Editorial Team