How many solutions does this system have?
y = x + 4
y - 4 = x

Since the first equation can be written: x + 4y = 4
and then multiplying it by 2 gives: 2x + 8y = 8 
That equation matches the 2nd one, so you really only have one equation with two unknowns. 
In this case, there will be an infinite number of solutions: 
x+8y = 8
y = (1/8)(-2x + 8)
y = -(1/4)x + 1
So the solutions are all the points on the line y = -(1/4)x + 1.