Simple question about solving equations and factorization

Question

I realize this is probably a really trivial question but it's been something I've been curious about for a while now. If I have the following equation:

$$ \\8x^7y^6(9-x-y)^2=x^8y^62(9-x-y) $$

and I want to simplify it, which method is correct, cancelling out the common terms like this:

$$ \\4(9-x-y)=x $$

Or, factorizing like this:

$$ \\8x^7y^6(9-x-y)^2-x^8y^62(9-x-y)=0 $$

$$ \\2x^7y^6(9-x-y)(36-5x-4y)=0 $$

Thanks!

Answer 1

Factorizing is the correct method, because cancelling out means you are dividing which is something you cannot do since $x,y$ or $(9-x-y)$ could be equal to zero.

After your factorization, you can easily see that $x$ or $y$ could be zero and $(9-x-y)$ can be zero too.

Answer 2

Factorizing is the correct method Because you do not know whether $x,y$ are 0 or not.

Answer 3

Both methods are valid but I personally like the second one better.

Because if you decide on the first one, though, you have to check that none of the terms you are dividing by is zero so you have to check a lot of cases.

With the second method you don't have to do this and you have all the solutions of the equations at first sight after the factorization (just set every factor equal to zero and solve).