* and / take priority over + and -

Let x=y

Then x^2 = xy [multiplied by x both sides]

Then x^2-y^2 = xy-y^2 [subtracted y^2 both sides]

Then (x+y)(x-y) = y(x-y) [factored each side]

Then x+y = y [cancelled (x-y) both sides]

Then x+x = x [substitute x in for y]

Then 2 = 1 [divide both sides by x]

A little induction on the above result shows that any natural number is equal to any other natural number

So there's no need to be arguing over the different answers given, as they are really just the same

Proof that women are evil

Women require time and money

Women = Time x Money

and we know that time is money, so Time = Money...

Women = Money x Money = Money^2

And we know that money is the root of all evil

Money = Sqrt (evil)

Therefore

Women = (Sqrt(evil^2))

So we conclude that

Women = Evil

Eldrick:
Then x+y = y [cancelled (x-y) both sides]

Implicitly this is dividing by (x-y) but as you've stated that x=y then this is tantamount to dividing by 0 - which is not allowed. Anything divided by 0 is considered Not Defined (this is not the same as infinity) - as another poster pointed out, it is an illegal operation.

Same argument holds for the post that followed that one, where the initial statement was a=0.

There are any number of supposed 'proofs' that 1=2 or 1=whatever, they almost all hinge on performing this illegal operation in some form or other which is usually dressed up algebraically to disguise the tactic.

As to the main point, BODMAS rules the order of operations, nowadays it is taught as BIDMAS (the I stands for Indices).

Hence 4+2x5 is 14, and brackets are not necessary to arrive at that - but they may help clarify things for people. When using a calculator you should be aware of these BODMAS rules as not all calculators automaticaly adhere to them. It's traditionally the cheapo calculators that use the simpler algorithm, no excuse for Microsoft though.

Other good ones that trip people up:
What's the square root of 1/4?
What's the first prime number?

correct moonshiner, the post from investor after was basically restating the argument in far fewer lines, but the division by 0 was a little more in your face then

why is the first prime number question tricky? i can only guess that some people like to bicker about whether 1 is a prime, i have it in my head that it isn't but it will just come down to a convention rather than any illuminating logical argument

I was taught that 2 is the first prime because, for reasons I've forgotten, 1 isn't a prime.