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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

user3482749
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Harry
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  • No, you need to prove the complete "or" condition. –  Dec 04 '18 at 17:31
  • It would, but you won't be able to, because it isn't true. For example, $x = 0, y = 5$ solves that equation, so whatever you've done to get $x = y$ is wrong. – user3482749 Dec 04 '18 at 17:32

2 Answers2

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

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

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

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Hint

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

mfl
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