Question

4x + 12 = -7y -y + 12 = 4x Solve with elimination

Answer 1

$\begin{cases}4x + 12 = -7y \\ -y + 12 = 4x \end{cases} \\ \\\begin{cases}4x + 7y = -12 \\ -4x-y =-12 \end{cases} \\+ ------- \\6y=-24\ \ /:6\\ \\y=-\frac{24}{6}\\ \\y=-4$

$4x+12=-7y\\ \\ 4x+12=-7*(-4)\\ \\4x=28-12\\ \\4x=16\ \ /:4\\ \\x=\frac{16}{4}\\ \\ x=4 \\ \\ \begin{cases}x=4 \\ y=-4 \end{cases}$

Answer 2

Take the second equation and flip it around so the y on the left ends up on the right and the 4x on the right ends up on the left.  This makes all negatives positive and all positives negative.   -4x   + 12  = y 

Then add the first equation to the second equation 
4x   +12   = -7y
-4x   + 12  = y    this eliminates the x's
          24 = - 6y            then divide by - 6
           - 6     - 6
         - 4    = y

So if you know that y = negative 4, you can substitute into either equation.  I pick the second one because I am a lazy person.  
  -y  + 12  = 4 x
-(-4) + 12 = 4 x     combine your numbers
     16      = 4 x        then divide by 4
       4      =  x
So your solution is: x = 4 and y = -4 or this is also written (4, -4)

Does that work for you?