Let's solve the system of equations using the elimination method. The given system is:

$1) \quad 2x + 2y = 6$ $2) \quad 3x - 5 = y$

Step 1: Substitute the expression for $y$ from equation (2) into equation (1)

From equation (2), we have:

$y = 3x - 5$

Substitute this into equation (1):

$2x + 2(3x - 5) = 6$

Step 2: Simplify the equation

Now, simplify the expression:

$2x + 6x - 10 = 6$ $8x - 10 = 6$

Step 3: Solve for $x$

Add 10 to both sides:

$8x = 16$

Now, divide both sides by 8:

$x = 2$

Step 4: Solve for $y$

Now that we have $x = 2$, substitute this back into equation (2) to find $y$:

$y = 3(2) - 5 = 6 - 5 = 1$

Final Solution:

The solution to the system of equations is: $x = 2, \quad y = 1$

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Would you like more details on the solution process, or do you have any questions?

Here are some related questions you can explore:

1. How do you use substitution to solve a system of equations?
2. What happens if the system of equations has no solution?
3. How do you solve a system of equations with three variables?
4. What are the differences between the elimination method and substitution method?
5. Can you solve a system of equations that has infinitely many solutions?

Tip: Always check your solution by substituting the values of $x$ and $y$ back into the original equations!