Question

`x+y+z=-1, 3x+y+4z=8,-x-y+7z=9`?

Answer 1

`x=3`
`y=-5`
`z=1`

Explanation:

There are three equations with three variables.

Make `y` the subject in all three equations:

`y= -x-z -1" "`..... equation 1
`y = -3x-4z+8" "` ... equation 2
`y= -x+7z-9" "`...equation 3

By equating the equations in pairs we can form two equations with the variables `x and z` and solve them simultaneously

Using equations 1 and 2: `" "y=y`

`" "-x-z-1 = -3x-4z+8`
`3x-x+4z-z=8+1" "larr` re-arrange

`2x+3z=9" "` equation A

Using equations 3 and 2 `" "y=y`

`" "-x+7z-9=-3x-4z+8" "larr` re-arrange

`3x-x+7z+4z=8+9`

`2x+11z=17" "` equation B

Now solve A and B for `x and z`

`" "2x+11z = 17color(white)(mmmmmmmmmmm)A`
`" "2x+3z =9color(white)(mmmmmmmmmmmm)B`

`A-B:" "8z = 8`
`color(white)(mmmmmm)z =1`

`2x +3(1)=9`
`2x +3=9`
`2x =6`
`x=3`

Now find `y` from equation 1
`y= -x-z -1`
`y = -(3)-(1)-1`
`y =-5`

Check with equation 2

`y =-3x-4z+8`
`y = -3(3)-4(1)+8`
`y=-9-4+8`
`y=-5`

Answer 2

`x=3`, `y=-5` and `z=1`

Explanation:

`x+y+z=-1`, `3x+y+4z=8` an `-x-y+7z=9`

From first equation, `z=-x-y-1`

Plug `z` into second an third ones;

`3x+y+4*(-x-y-1)=8`

`3x+y-4x-4y-4=8`

`-x-3y=12`

`-x-y+7*(-x-y-1)=9`

`-x-y-7x-7y-7=9`

`-8x-8y=16`

`-8*(x+y)=16` or `x+y=-2`

From second one, `x=-3y-12`

Plug `x` into third one;

`(-3y-12)+y=-2`

`-2y-12=-2`

`-2y=10`, so `y=-5`

Hence `x=-3y-12=(-3)*(-5)-12=3`

Thus, `z=-x-y-1=-3-(-5)-1=1`