`3x - 3y + 6z = 6, x + 2y - z = 5, 5x - 8y + 13z = 7` Solve the system of linear equations and check any solutions algebraically.

First, divide the first equation by 3 and obtain x-y+2z=2. Then express x=y-2z+2 and substitute it into the second and third equations:

y-2z+2+2y-z=5, or 3y-3z=3, or y-z=1,

and

5y-10z+10-8y+13z=7, or -3y+3z=-3, or y-z=1.

Thus there are only two independent equations and infinitely many solutions. They have a form z=z (any), y=z+1 and x=y-2z+2=z+1-2z+2=3-z. This is the answer.

Check the answer:

x-y+2z=3-z-z-1+2z=2 (true),

x+2y-z=3-z+2z+2-z=5 (true),

5x-8y+13z=15-5z-8z-8+13z=7(true).