We are given the system of equations:

$6x + 6y = 6$

$9x + 2y = 16$

Step 1: Solve for One Variable

To simplify, we can divide the first equation by 6:

$x + y = 1$

$9x + 2y = 16$

From the first equation, solve for $y$:

$y = 1 - x$

Step 2: Substitute into the Second Equation

Substituting $y = 1 - x$ into $9x + 2y = 16$:

$9x + 2(1 - x) = 16$

$9x + 2 - 2x = 16$

$7x + 2 = 16$

$7x = 14$

$x = 2$

Step 3: Solve for $y$

Using $y = 1 - x$:

$y = 1 - 2 = -1$

Solution:

$\boxed{(2, -1)}$

Would you like a detailed verification?

Related Questions:

1. How can this system be solved using the elimination method?
2. What happens if we change the second equation to $9x + 2y = 10$?
3. Can this system be solved graphically? How?
4. How do we determine if a system of equations has a unique solution, no solution, or infinitely many solutions?
5. How can we solve this system using matrix methods?

Tip: Always check your solution by substituting it back into the original equations!