The first set of steps solving 5x-y = 4 to get y = 5x-4 is correct.
The second set of steps are incorrect since it involves circular logic. Notice we're plugging y = 5x-4 back into the original equation we just solved. It would be like plugging an extension cord into itself, and somehow expecting electricity to magically come out of nowhere.
Instead we should plug that y expression into the first equation and solve for x like so:
8x+2y = -12
8x+2(5x-4) = -12
8x+10x-8 = -12
18x-8 = -12
18x = -12+8
18x = -4
x = -4/18
x = -2/9
Then we use this to find the value of y.
y = 5x-4
y = 5(-2/9)-4
y = -10/9-4
y = -10/9 - 36/9
y = (-10-36)/9
y = -46/9
In summary we have the solution (x,y) = ( -2/9, -46/9 )