Problem solving

Question

A student has proven that all numbers are equal! He shows you the following proof. What is wrong with this proof?

 

Pick any two numbers, a and b, and a non-zero number c that is the difference between a and b. Thus we have the following:

 

                  a=b+c        c≠ 0

Multiply both sides by a−b : a(a−b) = (b+c)(a−b)

which expands to:

a² −ab=ab−b² +ac−bc subtract ac from both sides:

a² −ab−ac=ab−b² −bc factor out a from the left side of the equation, and b from the right: a(a−b−c) =b(a−b−c)

Eliminate the common factor from both sides and we have: a=b

Answer 1

a-b is in fact equal to c, based on the original condition a+b=c. Therefore, a-b-c = c-c = 0. The final step of eleminating a-b-c from both sides thus necessitates dividing by zero, which is not possible.