why 2 =1 from the solution?

Question

let a=b
a^2=ab
a^2-b^2=ab-b^2
(a-b)(a+b)=b(a-b)
a+b=b
a+a=a
2a=a
2=1
why???????Is there any theory that i used is wrong.if it is correct,then is it 2 can also be equal to 1 in actual mathematics rules??

Answer 1

Proof that 2 equals 1

* Let a and b be equal non-zero quantities

a = b

* Multiply through by a

a2 = ab

* Subtract b2

a2 − b2 = ab − b2

* Factor both sides

(a − b)(a + b) = b(a − b)

* Divide out (a − b)

a + b = b

* Observing that a = b

b + b = b

* Combine like terms on the left

2b = b

* Divide by the non-zero b

2 = 1

Q.E.D.

The fallacy is in line 5: the progression from line 4 to line 5 involves division by (a − b), which is zero since a equals b. Since division by zero is undefined, the argument is invalid.

A variation:

* Let x and y be equal, non-zero quantities

x = y

* Add x to both sides

2x = x + y

* Take 2y from both sides

2x − 2y = x − y

* Factor out a two on the left side

2(x − y) = x − y

* Divide out (x − y)

2 = 1

Q.E.D.

The fallacy here is the same as above in that by dividing by (x − y), you are dividing by zero and as such, this argument is invalid. Also by this argument:

Answer 2

if a=b
then a-b=0
and you cannot cancel 0 from both sides as that means dividing both sides by zero which wud then result in infinite (anything divided by zero is infinite) therefore though this is a smart proof but has this critical mistake

Answer 3

The problem is that you are dividing both sides by zero (a-b=0).

Answer 4

In the forth line (a-b)(a+b)=b(a-b), (a-b)=0 therefore it is impossible to get to the next line as [(a-b)(a+b)]/0 is undefined.

Answer 5

error error error

cannot divide by (a-b)

reason = a-b = 0

fail fail fail

cannot divide by ZERO