\(\dfrac{x-1}{2}=\dfrac{y+3}{4}=\dfrac{z-5}{6}\Rightarrow\dfrac{5\left(z-5\right)}{30}=\dfrac{3\left(x-1\right)}{6}=\dfrac{4\left(y+3\right)}{16}=\dfrac{5\left(z-5\right)-3\left(x-1\right)-4\left(y+3\right)}{30-6-16}=\dfrac{5z-25-3x+3-4y-12}{8}=\dfrac{\left(5z-3x-4y\right)+\left(-25+3-12\right)}{8}=\dfrac{50+\left(-34\right)}{8}=\dfrac{16}{8}=2\)\(\Rightarrow\dfrac{x-1}{2}=2\Rightarrow x=5\\ \dfrac{y+3}{4}=4\Rightarrow y=13\\ \dfrac{z-5}{6}=2\Rightarrow z=17\)