simple proof algebra question on' or'

Question

If $x^2+5y=y^2+5x$ then $x=y$ or $x+y=5$, where $x$ and $y$ are real numbers . Prove this statement.

Can someone help me with this problem or how to approach it? I can get x=y Does this mean i have proved the statement because it is an 'or'? Thanks

Answer 1

Hint

$$x^2+5y=y^2+5x\iff y^2-5y+5x-x^2=0\iff y=\dfrac{5\pm |2x-5|}{2}.$$

Answer 2

$$x^2+5y=y^2+5x\iff x^2-y^2=5x-5y\iff(x+y)(x-y)=5(x-y) \\\iff(x+y-5)(x-y)=0.$$

Now a product is zero when either factor is zero, that means

$$x+y-5=0\text{ or }x-y=0.$$