[Math] Dividing one equation by another equation

Question

This is from Higher Algebra by Hall and Knight,

$u+v+\sqrt{uv}=39$…(1)

$u^2+v^2+uv=741$…(2)

we obtain by division

$u+v-\sqrt{uv}=19$

I don't know how do you divide one equation by another equation, can someone pls explain.

Answer 1

It's just like any other operation: if you have an equation that is true, and you do the same thing to both sides, you get another equation that is true.

For example, if we know $a=b$ and $c=d$ are both true, then we know $a+c = b+d$ is true: since $a$ and $b$ are the same thing, we've done the same thing to both sides of the equation $c+d$.

The only thing that differentiates division in this respect is that you have to pay a little more attention that the thing you do is not nonsense. i.e. dividing by $x$ is nonsense if you don't know $x$ is nonzero.