algebraic fallacy, 2 = 1

Using only basic algebraic manipulation, and trying to not skip any steps, I am going show that 2 = 1 .

a = b

we can do whatever we want to an equation as long as we do the same thing to both sides, so first add a

a + a = b + a

simplify

2a = b + a

subtract 2b

2a - 2b = a + b - 2b

group like terms

2a - 2b = a + (1-2)b

simplify

2a - 2b = a + (-1)b

adding a negative number is the same as subtracting

2a - 2b = a - b

factor out (a - b)

2(a - b) = 1(a - b)

divide by (a - b)

\frac{2(a - b)}{a-b} = \frac{1(a - b)}{a-b}

re-factor to isolate the one

2\left(\frac{a - b}{a-b}\right) = 1\left(\frac{a - b}{a-b}\right)

reduce the one

2(1) = 1(1)

simplify

2 = 1

where did it go wrong? The answer will be given in a later post.

My class was given this puzzle in high school one day, and I was not the first one to see the mistake, but it was quite memorable.