### Brain Teasers

# 2=1?

Read this following proof carefully.

X=Y Given

X^2=XY Multiply both sides by X

X^2-Y^2=XY-Y^2 Subtract Y^2 from both sides

(X+Y)(X-Y)=Y(X-Y) Factor both sides

(X+Y)=Y Cancel out common factors

Y+Y = Y Substitute in from line (1)

2Y = Y Collect the Y's

2 = 1 Divide both sides by Y

Now obviously, 2 doesn't equal 1. So what's wrong with this proof?

X=Y Given

X^2=XY Multiply both sides by X

X^2-Y^2=XY-Y^2 Subtract Y^2 from both sides

(X+Y)(X-Y)=Y(X-Y) Factor both sides

(X+Y)=Y Cancel out common factors

Y+Y = Y Substitute in from line (1)

2Y = Y Collect the Y's

2 = 1 Divide both sides by Y

Now obviously, 2 doesn't equal 1. So what's wrong with this proof?

### Hint

It's in lines 4 and 5.### Answer

'Cancelling' the common factors from line (4) to (5) means dividing by the factor (X-Y). Since X=Y, this is a division by 0, the results of which are undefined.Hide Hint Show Hint Hide Answer Show Answer

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