The way I learned it a dog’s age ago is to think of the equation as being like a scale; What you do to one side of the = sign, you have to do the other side. When you are trying to get the variable (the letter for the unknown term on one side), you need to do what you have to do to make zero on the side of the equation that the variable is on. If you have a negative number, you have to add the same number. Because you need to balance, you need to do it on both sides of the equation. A positive number plus the negative of that number always equals zero.
In your example, y – 15 = -7, to get y by itself you have to add 15 to each side, because
-15 +15 = 0, and this puts y by itself (y-0 = y)
Because you need to keep the equation balanced, you need to also add 15 to the other side,
or -7 + 15, which equals 8, or y = 8.