Let's solve the system of equations using the substitution method:

From the third equation, we can solve for y in terms of z:
2y + 7z = -6
2y = -7z - 6
y = -7z/2 - 3

Now substitute this expression for y into the first equation:
2x - (-7z/2 - 3) - 3z = 5
2x + 7z/2 + 3 - 3z = 5
2x + 7z/2 - 3z = 2
4x + 7z - 6z = 4
4x + z = 4

Now substitute for y again in the second equation:
3x + 4(-7z/2 - 3) - 5z = -9
3x - 14z - 12 - 5z = -9
3x - 19z - 12 = -9
3x - 19z = 3
x = (19z + 3) / 3
x = 6z + 1

Now we have the expressions for x, y, and z:
x = 6z + 1
y = -7z/2 - 3
z = z

We can now solve for z by substituting x and y in the third equation:
2(-7z/2 - 3) + 7z = -6
-7z - 6 + 7z = -6
-6 = -6

Since this equation is true, z can be any value. By substituting back into x and y, we can find the values for x and y accordingly.