Đáp án:

$\begin{array}{l}
a)\dfrac{{x - 2}}{5} = \dfrac{3}{8}\\
 \Rightarrow x - 2 = 5.\dfrac{3}{8}\\
 \Rightarrow x - 2 = \dfrac{{15}}{8}\\
 \Rightarrow x = \dfrac{{15}}{8} + 2\\
 \Rightarrow x = \dfrac{{15}}{8} + \dfrac{{16}}{8}\\
 \Rightarrow x = \dfrac{{31}}{8}\\
Vậy\,x = \dfrac{{31}}{8}\\
b)\dfrac{{x - 1}}{{x - 5}} = \dfrac{6}{7}\\
 \Rightarrow 7.\left( {x - 1} \right) = 6.\left( {x - 5} \right)\\
 \Rightarrow 7x - 7 = 6x - 30\\
 \Rightarrow 7x - 6x =  - 30 + 7\\
 \Rightarrow x =  - 23\\
Vậy\,x =  - 23\\
c)\dfrac{{{x^2}}}{6} = \dfrac{{24}}{{25}}\\
 \Rightarrow {x^2} = \dfrac{{24}}{{25}}.6\\
 \Rightarrow {x^2} = \dfrac{{144}}{{25}}\\
 \Rightarrow \left[ \begin{array}{l}
x = \dfrac{{12}}{5}\\
x =  - \dfrac{{12}}{5}
\end{array} \right.\\
Vậy\,x = \dfrac{{12}}{5};x =  - \dfrac{{12}}{5}\\
d)\dfrac{x}{y} = \dfrac{7}{{13}}\\
 \Rightarrow \dfrac{x}{7} = \dfrac{y}{{13}} = \dfrac{{x + y}}{{7 + 13}} = \dfrac{{ - 60}}{{20}} =  - 3\\
 \Rightarrow \left\{ \begin{array}{l}
x =  - 21\\
y =  - 39
\end{array} \right.\\
Vậy\,x =  - 21;y =  - 39\\
e)\dfrac{x}{{19}} = \dfrac{y}{{21}}\\
 = \dfrac{{2x}}{{38}} = \dfrac{{2x - y}}{{38 - 21}} = \dfrac{{34}}{{17}} = 2\\
 \Rightarrow \left\{ \begin{array}{l}
x = 38\\
y = 42
\end{array} \right.\\
Vậy\,x = 38;y = 42\\
f)\dfrac{x}{3} = \dfrac{y}{4} = k \Rightarrow \left\{ \begin{array}{l}
x = 3k\\
y = 4k
\end{array} \right.\\
{x^2} + {y^2} = 100\\
 \Rightarrow {\left( {3k} \right)^2} + {\left( {4k} \right)^2} = 100\\
 \Rightarrow 25{k^2} = 100\\
 \Rightarrow {k^2} = 4\\
 \Rightarrow \left[ \begin{array}{l}
k = 2\\
k =  - 2
\end{array} \right.\\
 \Rightarrow \left[ \begin{array}{l}
x = 6;y = 8\\
x =  - 6;y =  - 8
\end{array} \right.\\
Vậy\,x =  \pm 6;y =  \pm 8;
\end{array}$