Question

How do you solve `-2x - 4y + 2z = -8`, `-x - 5y +12z = 5`, `3x + 5y - z = 10`?

Answer 1

`x=-9` and `y=8` and `z=3`

Explanation:

From the given equations
`-2x-4y+2z=-8" " "`first equation
`-x-5y+12z=5" " "`second equation
`3x+5y-z=10" " "`third equation

We eliminate the variable x first

multiply the terms of the second equation by 2 then subtract it from the first equation

`-2x-4y+2z=-8" " "`first equation
`-2x-10y+24z=10" " "`second equation
after subtraction
`6y-22z=-18` fourth equation

multiply the original second equation by 3 then add to the third equation

`-3x-15y+36z=15" " "`second equation
`3x+5y-z=10" " "`third equation

after addition
`-10y+35z=25" " "` fifth equation reducible to
`-2y+7z=5larr`reduced fifth equation
Multiply this fifth equation by 3 then add to the fourth equation

`-6y+21z=15" " "`fifth equation
`6y-22z=-18` fourth equation

the result is
`-z=-3`
and
`z=3`

Using the reduced fifth equation `-2y+7z=5` and `z=3` solve for y

`-2y+7z=5`
`-2y+7(3)=5`
`-2y+21=5`
`-2y=-16`
`y=8`

Using the original second equation `-x-5y+12z=5" "`and `y=8` and `z=3` solve for `x`

`-x-5y+12z=5" "`
`-x-5(8)+12(3)=5" "`
`-x-40+36=5`
`-x=9`
`x=-9`

the solution is `x=-9` and `y=8` and `z=3`

God bless...I hope the explanation is useful..