Let's start solving the system of equations.

The first equation is:
3x + 6y - y = 27
Simplify this equation:
3x + 5y = 27

The second equation is:
4x + 4y - 3x = 23
Simplify this equation:
x + 4y = 23

Now we have the following system of equations:
3x + 5y = 27
x + 4y = 23

To solve this system, we can use the substitution method or elimination method. Let's use the substitution method.

From the second equation, we have: x = 23 - 4y

Substitute x = 23 - 4y into the first equation:
3(23 - 4y) + 5y = 27
69 - 12y + 5y = 27
69 - 7y = 27
-7y = -42
y = 6

Now that we have found the value of y, substitute y back into the second equation to find x:
x + 4(6) = 23
x + 24 = 23
x = -1

Therefore, the solution to the system of equations is x = -1 and y = 6.