To solve this system of equations, we can use the method of substitution or elimination.

First, let's solve the first equation for x in terms of y and z:

x = 6 + y - 7z

Now, substitute this expression for x into the other two equations:

2(6 + y - 7z) + 3y - 3z = 10
12 + 2y - 14z + 3y - 3z = 10
5y - 17z = -2

3(6 + y - 7z) + 2y + 5z = 17
18 + 3y - 21z + 2y + 5z = 17
5y - 16z = -1

Now, we have a system of two equations:

5y - 17z = -2
5y - 16z = -1

Subtracting the second equation from the first, we get:

-16z - (-17z) = -1 - (-2)
z = 1

Now, substitute z = 1 back into one of the original equations:

5y - 17(1) = -2
5y - 17 = -2
5y = 15
y = 3

Finally, substitute y = 3 and z = 1 back into x = 6 + y - 7z:

x = 6 + 3 - 7(1)
x = 2

Therefore, the solution to the system of equations is x = 2, y = 3, z = 1.