Another way of looking at it is that to solve for z you must "undo" what has been done to z. Initially, x= az+ b. z is not "isolated" (alone on one side of the equation) because two things have been done to it: (1) multiply by a, (2) add b. To undo, you must do the opposite of each step- in the opposite order. That is, instead of "multiply by a and then add b" we must "subtract b and then divide by a". And, of course, anything we do to one side of an equation, we must do to the other.
Starting from x= az+ b, subtracting b from both sides gives x-b= az and then dividing both sides by a gives (x-b)/a= z.