Chapter 6 - Section 6.6 - Rational Equations and Problem Solving - Exercise Set - Page 389: 63

$x=2$

By cross-multiplication and using the properties of equality, the solution to the given equation, $ \dfrac{x-3}{2}=\dfrac{x-5}{6} ,$ is \begin{array}{l} 6(x-3)=2(x-5) \\\\ 6x-18=2x-10 \\\\ 6x-2x=-10+18 \\\\ 4x=8 \\\\ x=\dfrac{8}{4} \\\\ x=2 .\end{array}