From the give equations
Let
5x+3y-2z=-12" "first equation
x+7y-9z=7" "second equation
x-y+z=-4" "third equation
Also the equations are equivalent to the following; after multiplying the terms of second and third equations by 5
5x+3y-2z=-12" "first equation
5x+35y-45z=35" "second equation
5x-5y+5z=-20" "third equation
Now we eliminate x using the first and second equations by subtraction
5x+3y-2z=-12" "first equation
underline(5x+35y-45z=35" ")second equation
0-32y+43z=47
32y-43z=47" "fourth equation
Now we eliminate x using the first and third equations by subtraction
5x+3y-2z=-12" "first equation
underline(5x-5y+5z=-20" ")third equation
0+8y-7z=8
8y-7z=8" "fifth equation
Multiply the fifth equation by 4 then subtract the fourth equation
32y-28z=32" "fifth equation
underline(32y-43z=47" ")fourth equation
0+15z=-15
15z=-15
color(red)(z=-1)
Use the 8y-7z=8" "fifth equation and z=-1 to solve for y
8y-7z=8" "fifth equation
8y-7(-1)=8" "fifth equation
8y+7=8
8y=8-7
8y=1
y=1/8
color(red)(y=1/8)
Use the third equation x-y+z=-4" " and color(red)(z=-1) and color(red)(y=1/8) to solve for x
x-y+z=-4" "third equation
x-(1/8)+(-1)=-4" "third equation
x=1/8+1-4
x=(1-24)/8
color(red)(x=-23/8)
God bless...I hope the explanation is useful.