Why do we have to do the same things to both sides of an equation?

Solution 1:

It comes down to what the equals sign means. It means, as @MaliceVidrine says, that the two things mentioned on either side of the equation are the same thing. If you do something to the left side, you have presumably changed it. If you don't do the same thing to the right side, you are now asserting that two different things are the same, and you have told a lie.

Solution 2:

If $ a = b $ in a set $ S $ and $ f: S \to T $ is a function, then $ f (a)=f (b) $ (this is the definition of a function). Doing the same thing to both sides of an equation in, say, a ring, is a special case of this. (e.g. there is a set function $ R \to R $ given by $r \mapsto r + 3$ and adding 3 to both sides of an equation is applying this function.

Solution 3:

This may sound a little daft, but I seldom see this issue explained in what I think is the simplest way: you have to apply any operation to both sides of an equation because it's the same entity being referred to on both sides. For some anonymous function $f$ and $x,y$ such that $x=y$, to apply $f$ to $x$ and not to $y$ is to apply $f$ to $x$ and also not apply $f$ to $x$ (since $x=y$).

You can, in practice, actually end up with $f(x)=y\wedge x=y$, but what this means is that $f(x)=x$ (that is, $x$ is a fixed point of $f$). If we were able to pass from an equality to a one-sided application of a function as a general rule, we would be accepting as a general principle that every point is a fixed point of every function.