\(\begin{array}{l} a) \Leftrightarrow \frac{{{3^x}}}{3} + 4.\frac{{{3^x}}}{{{3^2}}} = \frac{7}{{243}}\\ \Leftrightarrow {3^x}\left( {\frac{1}{3} + \frac{4}{9}} \right) = \frac{7}{{243}}\\ \Leftrightarrow {3^x}.\frac{7}{9} = \frac{7}{{243}}\\ \Leftrightarrow {3^x} = \frac{7}{{243}}:\frac{7}{9}\\ \Leftrightarrow {3^x} = \frac{1}{{27}}\\ \Leftrightarrow {3^x} = {3^{ - 2}} \Leftrightarrow x = - 2\\ b)\, \Rightarrow \frac{{x\left( {x - y} \right)}}{{y\left( {x - y} \right)}} = \frac{3}{{10}}:\left( { - \frac{3}{{50}}} \right) = - 5\\ \Leftrightarrow \frac{x}{y} = - 5 \Rightarrow x = - 5y\\ \Rightarrow \left( { - 5y} \right)\left( { - 5y - y} \right) = \frac{3}{{10}}\\ \Leftrightarrow 30{y^2} = \frac{3}{{10}} \Leftrightarrow {y^2} = \frac{1}{{100}} \Rightarrow y = \frac{1}{{10}} \Rightarrow x = - \frac{1}{2}\\ c)\,\left\{ \begin{array}{l} {\left( {7x - 5y} \right)^{2018}} \ge 0\\ {\left( {3x - 2z} \right)^{2020}} \ge 0\\ {\left( {xy + yz + xz - 4500} \right)^{2022}} \ge 0 \end{array} \right.\\ \Rightarrow {\left( {7x - 5y} \right)^{2018}} + {\left( {3x - 2z} \right)^{2020}} + {\left( {xy + yz + xz - 4500} \right)^{2022}} = 0\\ \Leftrightarrow \left\{ \begin{array}{l} {\left( {7x - 5y} \right)^{2018}} = 0\\ {\left( {3x - 2z} \right)^{2020}} = 0\\ {\left( {xy + yz + xz - 4500} \right)^{2022}} = 0 \end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l} 7x = 5y\\ 3x = 2z\\ xy + yz + xz - 4500 = 0 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} y = \frac{{7x}}{5}\\ z = \frac{{3x}}{2}\\ xy + yz + xz - 4500 = 0 \end{array} \right.\\ \Rightarrow x.\frac{{7x}}{5} + \frac{{7x}}{5}.\frac{{3x}}{2} + x.\frac{{3x}}{2} = 4500\\ \Leftrightarrow {x^2}.5 = 4500 \Leftrightarrow {x^2} = 900\\ \Leftrightarrow \left[ \begin{array}{l} x = 30 \Rightarrow y = 42;z = 45\\ x = - 30 \Rightarrow y = - 42;z = - 45 \end{array} \right. \end{array}\)