Question

How do you solve x+3y-z=13, 2x-5z=23, 4x-y-2z=14?

Answer 1

`x=366/114`

`y=738/342`

`z=(-189)/57`

Explanation:

`x+3y-z=13` ------------------(1)
`2x-5z=23`------------------(2)
`4x-y-2z=14` ----------------(3)

Solve equations (1) and (2)

`x+3y-z=13` ------------------(1)
`2x-5z=23`------------------(2)

Multiply equation (1) with `2` to eliminate `x` term

`x+3y-z=13 xx 2` ------------------(1)
`2x+0y-5z=23`------------------(2)

`2x+6y-2z=26` ------------------(1) Subtract (2) from (1)
`2x+0y-5z=23`------------------(2)
`6y+3z=3` ----------------(4)

Solve equations (2) and (3) and eliminate `x` term

`2x+0y-5z=23`------------------(2)
`4x-y-2z=14` ----------------(3)

Multiply (2) with `2`

`2x+0y-5z=23 xx 2`------------------(2)
`4x-y-2z=14` ----------------(3)

`4x+0y-10z=46`------------------(2) Subtract (3) from (2)
`4x-y-2z=14` ----------------(3)
`y-8z=32` ----------------(5)

Take equations (4) and (5)

`6y+3z=3` ----------------(4)
`y-8z=32` ----------------(5)

Multiply equation (5) with `6`

`6y+3z=3` ----------------(4) Subtract (5) from (4)
`6y-54z=192` ----------------(5)
`57z=-189`

`z=(-189)/57`

Substitute the value of `z` in equation (2)

`2x-5z=23`------------------(2)
`2x-5((-189)/57)=23`------------------(2)
`2x+945/57=23`------------------(2) Multiply both sides by `57`
`114x+945=1311`-------- Solve it for `x`
`114x=1311-945=366`

`x=366/114`

Substitute the value of `x` and `z` in equation (1)

`x+3y-z=13` ------------------(1)
`366/114+3y-(-189)/57=13`
`366/114+3y+189/57=13` ---- Multiply both sides with `114`
`366+342y+378=1482`-------solve it for `y`

`744+342y=1482`
`342y=1482-744=738`

`y=738/342`