Question

Solve $\frac{3x-4}{7}+\frac{7}{3x-4}=\frac{5}{2}$

Answer 1

The given equation is $\frac{3x-4}{7}+\frac{7}{3x-4}=\frac{5}{2}$

Solving the above equation

$$\begin{gathered} \frac{3\mathrm{x}-4}{7}+\frac{7}{3\mathrm{x}-4}=\frac{5}{2} \\ \mathrm{Consider}\frac{3\mathrm{x}-4}{7}=\mathrm{y} \\ \Rightarrow \mathrm{y}+\frac{1}{\mathrm{y}}=\frac{5}{2} \\ \Rightarrow 2{\mathrm{y}}^2-5\mathrm{y}+2=0 \\ \Rightarrow 2{\mathrm{y}}^2-4\mathrm{y}-\mathrm{y}+2=0 \\ \Rightarrow 2\mathrm{y}(\mathrm{y}-2)-1(\mathrm{y}-2)=0 \\ \Rightarrow (2\mathrm{y}-1)(\mathrm{y}-2)=0 \\ \Rightarrow \mathrm{either}(2\mathrm{y}-1)=0\mathrm{or}(\mathrm{y}-2)=0 \\ \Rightarrow \mathrm{y}=\frac{1}{2},2 \\ \Rightarrow \mathrm{When}\mathrm{y}=\frac{1}{2},\frac{3\mathrm{x}-4}{7}=\frac{1}{2} \\ \Rightarrow 6\mathrm{x}-8=7 \\ \Rightarrow \mathrm{x}=\frac{15}{6}=\frac{5}{2} \\ \mathrm{When}\mathrm{y}=2 \\ \Rightarrow \frac{3\mathrm{x}-4}{7}=2 \\ \Rightarrow 3\mathrm{x}=18 \\ \Rightarrow \mathrm{x}=6 \end{gathered}$$

Therefore, the required values are $x=\frac{5}{2},6$