Chapter 3.5, Problem 5EE

Algebraic arguments often consist of a list of statements. In a valid algebraic argument, each statement (after the premises) can be deduced from the previous statements. The following argument that $1=2$ contains exactly one step that is not valid. Identify the step and explain why it is not valid.

Let

$\begin{array}{l}a=b\text{Premise}.\\ a^2=ab\text{Multiplyeach side by }a.\\ a^2-b^2=ab-b^2\text{Subtract }{b}^2\text{ from each side}\text{.}\\ \left(a+b\right)\left(a-b\right)=b\left(a-b\right)\text{Factor each side}\text{.}\\ a+b=b\text{Divide each side by }(a-b).\\ b+b=b\text{Substitute }b\text{ for }a.\\ 2b=b\text{Collect like terms}\text{.}\\ 2=1\text{Divide each side by }b.\end{array}$