Another way to illustrate why division by $0$ is undefined is to study the graph of $y=\frac{1}{x}.$ By using a table of values some points on the curve can be found.

Graphing the points on a coordinate plane allows the behavior of the function to be shown. The red points show what happens as $x$ approaches $0$ from the left, and the green points show what happens as $x$ approaches $0$ from the right.

The closer $x$ comes to $0,$ on both sides, the more extreme the values of the functions become. The quotient approaches $+\infty$ when $x$ approaches $0$ from the right, and $\text{-}\infty$ when $x$ approaches from the left. Because $f(x)=\frac{1}{x}$ is a function, $f$ can only produce one value when $x=0.$ Therefore, division by $0$ is undefined.

Since both sides of the equation are divided by $a-b=0,$ and $a-b=0,$ the expression was validly manipulated to create a false statement.