Đáp án:

Áp dụng tính chất dãy tỉ số bằng nhau ta có

$\begin{array}{l}
1)\dfrac{x}{y} = \dfrac{9}{{10}}\\
 \Rightarrow \dfrac{x}{9} = \dfrac{y}{{10}} = \dfrac{{y - x}}{{10 - 9}} = \dfrac{{12}}{1} = 12\\
 \Rightarrow \left\{ \begin{array}{l}
x = 12.9 = 108\\
y = 12.10 = 120
\end{array} \right.\\
\text{Vậy}\,x = 108;y = 120\\
2)\\
2x = 3y = 4z\\
 \Rightarrow \dfrac{{2x}}{{12}} = \dfrac{{3y}}{{12}} = \dfrac{{4z}}{{12}}\\
 = \dfrac{x}{6} = \dfrac{y}{4} = \dfrac{z}{3} = \dfrac{{x + y + z}}{{6 + 4 + 3}} = \dfrac{{39}}{{13}} = 3\\
 \Rightarrow \left\{ \begin{array}{l}
x = 18\\
y = 12\\
z = 9
\end{array} \right.\\
3)2x = 3y;5y = 7z\\
 \Rightarrow \left\{ \begin{array}{l}
y = \dfrac{2}{3}x\\
y = \dfrac{7}{5}z
\end{array} \right.\\
 \Rightarrow \dfrac{2}{3}x = y = \dfrac{7}{5}z\\
 \Rightarrow \dfrac{{2x}}{{3.14}} = \dfrac{y}{{14}} = \dfrac{{7z}}{{5.14}}\\
 \Rightarrow \dfrac{x}{{21}} = \dfrac{y}{{14}} = \dfrac{z}{{10}}\\
 = \dfrac{{3x}}{{63}} = \dfrac{{7y}}{{98}} = \dfrac{{5z}}{{50}} = \dfrac{{3x - 7y + 5z}}{{63 - 98 + 50}}\\
3x - 7y + 5z = ???\\
4)\dfrac{{x - 1}}{2} = \dfrac{{y - 2}}{3} = \dfrac{{z - 3}}{4}\\
 = \dfrac{{2y - 4}}{6} = \dfrac{{3z - 9}}{{12}}\\
 = \dfrac{{x - \left( {2y - 4} \right) + \left( {3z - 9} \right)}}{{2 - 6 + 12}}\\
 = \dfrac{{x - 2y + 3z + 4 - 9}}{8}\\
 = \dfrac{{14 + 4 - 9}}{8} = \dfrac{9}{8}\\
 \Rightarrow \left\{ \begin{array}{l}
x - 1 = \dfrac{9}{8}.2 = \dfrac{9}{4} \Rightarrow x = \dfrac{{13}}{4}\\
y - 2 = \dfrac{9}{8}.3 = \dfrac{{27}}{8} \Rightarrow y = \dfrac{{43}}{8}\\
z - 3 = \dfrac{9}{8}.4 = \dfrac{9}{2} \Rightarrow z = \dfrac{{15}}{2}
\end{array} \right.\\
\text{Vậy}\,x = \dfrac{{13}}{4};y = \dfrac{{43}}{8};z = \dfrac{{15}}{2}
\end{array}$