You have provided 3 expressions involving 3 variables, but no equations, so we can make the expressions into equations by supplying three constants:
6x-3y+2z=A, -3x+2y-z=B, 5x+6y-7z=C
Double the second equation: -6x+4y-2z=2B and we can see how to find y using the vertical method by adding this to the first equation:
We need two equations to find the other two variables, so substitute y=A+2B in the original 2nd and 3rd equations:
-3x+2A+4B-z=B, -3x-z=B-2A-4B=-2A-3B, z=-3x+2A+3B
Now substitute for z:
We can solve for x:
In terms of the constants A, B and C the solution is:
x=(8A+9B+C)/26, y=A+2B, z=(28A+51B-3C)/26.
We can put in some values for the constants as an example. Let A=B=1 and C=9, then:
x=1, y=3, z=2.