Đáp án:

$a)\left\{\begin{array}{l} x=7\\y=4\end{array} \right.\\ b)\left\{\begin{array}{l} x=\dfrac{41}{8}\\ y=-\dfrac{235}{23}\end{array} \right.\\ c) \left\{\begin{array}{l} y=-2\\ x=\dfrac{1}{2}\end{array} \right.\\ d)\left\{\begin{array}{l} x=\dfrac{8}{3}\\ y=\dfrac{17}{6}\end{array} \right.\\ e)\left\{\begin{array}{l} y=16\\ x=9\end{array} \right.$

$f)$Hệ vô nghiệm.

Giải thích các bước giải:

$a)\left\{\begin{array}{l} \dfrac{4}{x-3}+\dfrac{5}{y+1}=2\\ \dfrac{5}{x-3}+\dfrac{1}{y+1}=\dfrac{29}{20}\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} \dfrac{4}{x-3}+\dfrac{5}{y+1}=2\\ \dfrac{25}{x-3}+\dfrac{5}{y+1}=\dfrac{29}{4}\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l}+\dfrac{5}{y+1}=2-\dfrac{4}{x-3}\\ \dfrac{21}{x-3}=\dfrac{21}{4}\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} x=7\\\dfrac{5}{y+1}=1\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} x=7\\y=4\end{array} \right.\\ b)\left\{\begin{array}{l} \dfrac{8}{x}-\dfrac{1}{y+12}=1\\ \dfrac{1}{x}+\dfrac{5}{y+12}=3\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} \dfrac{8}{x}-\dfrac{1}{y+12}=1\\ \dfrac{8}{x}+\dfrac{40}{y+12}=24\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} \dfrac{8}{x}-\dfrac{1}{y+12}=1\\ \dfrac{41}{y+12}=23\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} \dfrac{8}{x}=\dfrac{1}{y+12}+1\\ y=-\dfrac{235}{23}\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} \dfrac{8}{x}=\dfrac{64}{41}\\ y=-\dfrac{235}{23}\end{array} \right.\\\Leftrightarrow \left\{\begin{array}{l} x=\dfrac{41}{8}\\ y=-\dfrac{235}{23}\end{array} \right.\\ c)\left\{\begin{array}{l} \dfrac{4}{2x+1}+\dfrac{9}{y-1}=-1\\ \dfrac{3}{2x+1}-\dfrac{2}{y-1}=\dfrac{13}{6}\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} \dfrac{12}{2x+1}+\dfrac{27}{y-1}=-3\\ \dfrac{12}{2x+1}-\dfrac{8}{y-1}=\dfrac{26}{3}\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} \dfrac{35}{y-1}=- \dfrac{35}{3}\\ \dfrac{12}{2x+1}=\dfrac{8}{y-1}+\dfrac{26}{3}\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} y=-2\\ \dfrac{12}{2x+1}=6\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} y=-2\\ x=\dfrac{1}{2}\end{array} \right.\\ d)\left\{\begin{array}{l} \dfrac{1}{x-1}+\dfrac{1}{y-2}=2\\ \dfrac{2}{y-2}-\dfrac{3}{x-1}=1\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} \dfrac{3}{x-1}+\dfrac{3}{y-2}=6\\ \dfrac{2}{y-2}-\dfrac{3}{x-1}=1\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} \dfrac{3}{x-1}+\dfrac{3}{y-2}=6\\ \dfrac{5}{y-2}=7\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} \dfrac{3}{x-1}=6-\dfrac{3}{y-2}\\ y=\dfrac{19}{7}\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} \dfrac{3}{x-1}=\dfrac{9}{5}\\ y=\dfrac{17}{6}\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} x=\dfrac{8}{3}\\ y=\dfrac{17}{6}\end{array} \right.\\ e)\left\{\begin{array}{l} 3\sqrt{x}-\sqrt{y}=5\\ 2\sqrt{x}+3\sqrt{y}=18\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} 9\sqrt{x}-3\sqrt{y}=15\\ 2\sqrt{x}+3\sqrt{y}=18\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} 9\sqrt{x}-3\sqrt{y}=15\\ 11\sqrt{x}=33\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} 3\sqrt{y}=9\sqrt{x}-15\\ x=9\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} y=16\\ x=9\end{array} \right.\\ f)\left\{\begin{array}{l} 3\sqrt{x}+2\sqrt{y}=6\\ \sqrt{x}-\sqrt{y}=4,5\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} 3\sqrt{x}+2\sqrt{y}=6\\ 2\sqrt{x}-2\sqrt{y}=9\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} 3\sqrt{x}+2\sqrt{y}=6\\ 5\sqrt{x}=15\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} 2\sqrt{y}=6-3\sqrt{x}\\ x=9\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} 2\sqrt{y}=-\dfrac{3}{2}\\ x=9\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} y \in \varnothing\\ x=9\end{array} \right.$

$\Leftrightarrow$ Hệ vô nghiệm.