The error arises in lines (4) and (5).
Because X = Y, the common factor (X-Y) is equal to zero. Therefore by cancelling out the common factor, one is dividing both sides by zero, the results of which are undefined. The result of this calculation can therefore be seen to be a simple fallacy.
| (1) X = Y | Given |
| (2) X2 = XY | Multiply both sides by X |
| (3) X2 - Y2 = XY - Y2 | Subtract Y2 from both sides |
| (4) (X+Y)(X-Y) = Y(X-Y) | Factoring both sides by (x-y), ie zero, is nonsensical |
| (5) (X+Y) = Y | Dividing both sides by zero leads to undefined results |
| (6) Y+Y = Y | Substitute in from line (1) |
| (7) 2Y = Y | Collect the Y's |
| (8) 2 = 1 | Divide both sides by Y |
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