If $\frac{a}{b}=\frac{x}{y}$, is $\frac{x-a}{y-b}=\frac{x}{y}$?

Question

Does this hold? $b,y \neq 0$, $b \neq y$.

Answer 1

So by $\frac ab = \frac xy$ we have $x = ka$, $y = kb$ for some $k \ne 0,1$. This gives $$ \frac{x-a}{y-b} = \frac{ka - a}{kb- b} = \frac{a(k-1)}{b(k-1)} = \frac ab = \frac xy. $$

Answer 2

If $\dfrac{a}{b}=\dfrac{x}{y}$ then $ay=xb$ so $xy-ay=xy-xb$, $y(x-a)=x(y-b) \to \dfrac{x-a}{y-b}=\dfrac{x}{y}$

Answer 3

\begin{align} \frac{a}{b}&=\frac{x}{y}\\ \frac{x}{a}&=\frac{y}{b}\\ \frac{x}{a}-1&=\frac{y}{b}-1\\ \frac{x-a}{a}&=\frac{y-b}{b}\\ \frac{x-a}{y-b}&=\frac{a}{b}\\ \frac{x-a}{y-b}&=\frac{x}{y} \end{align}