The given system of equations is:

1. $9u - y + 2 = 7$
2. $10u - 2y + 2 = 9$
3. $6u - 2y + 2 = 7$

Let's simplify these equations.

Step 1: Simplifying each equation

1. $9u - y + 2 = 7$ simplifies to $9u - y = 5$.
2. $10u - 2y + 2 = 9$ simplifies to $10u - 2y = 7$.
3. $6u - 2y + 2 = 7$ simplifies to $6u - 2y = 5$.

Now we have: $9u - y = 5$ $10u - 2y = 7$ $6u - 2y = 5$

Step 2: Solving the system

Subtract the third equation from the second: $(10u - 2y) - (6u - 2y) = 7 - 5$ This simplifies to: $4u = 2 \implies u = \frac{2}{4} = \frac{1}{2}$

Step 3: Substitute $u = \frac{1}{2}$ into one of the equations

Using $9u - y = 5$: $9\left(\frac{1}{2}\right) - y = 5$ $\frac{9}{2} - y = 5$ $y = \frac{9}{2} - 5 = \frac{9}{2} - \frac{10}{2} = -\frac{1}{2}$

Thus, the solution is: $u = \frac{1}{2}, \quad y = -\frac{1}{2}$

Would you like further details or have any questions about these steps?

Here are five related questions to expand your understanding:

1. How would you solve this system if you used substitution instead of elimination?
2. Can you solve this system using matrix methods, like Gaussian elimination?
3. How would you check if this system has a unique solution, no solution, or infinitely many solutions?
4. What happens if one of the equations is inconsistent with the others?
5. How would you solve a similar system with three variables?

Tip: Always simplify equations as much as possible to make the solving process easier.