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:

6x-3y+2z=A +

-6x+4y-2z=2B

y=A+2B

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

5x+6A+12B-7z=C, 5x-7z=C-6A-12B.

Now substitute for z:

5x-7(-3x+2A+3B)=C-6A-12B.

We can solve for x:

5x+21x=C-6A-12B+14A+21B=C+8A+9B.

26x=C+8A+9B, x=(C+8A+9B)/26.

And z=-3x+2A+3B=

-3(C+8A+9B)/26+2A+3B=(-3C-24A-27B+52A+78B)/26,

z=(28A+51B-3C)/26.

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.