Question

Solve the following equations. $\left({7.3}^{x+1}-5^{x+2}\right)=(3^{x+4}-5^{x+3})$

Answer 1

$7.3^{x+1}-5^{x+2}=3^{x+4}-5^{x+3}$

$\Rightarrow 7\cdot 3^x\cdot 3-5^x\cdot 5^2=3^x\cdot 3^4-5^x\cdot 5^3$

$\Rightarrow 21\cdot 3^x-5^x\cdot 25=3^x\cdot 81-5^x\cdot 125$

$21\cdot 3^x-3^x\cdot 81=5^x\cdot 25-5^x\cdot 125$

$\Rightarrow 3^x(21-18)=5^x(25-125)$

$\Rightarrow \frac{3^x}{5^x}=\frac{100}{60}$

$\Rightarrow {\left(\frac{3}{5}\right)}^x=\frac{5}{3}$

$\Rightarrow {\left(\frac{3}{5}\right)}^x={\left(\frac{3}{5}\right)}^{-1}$

$\therefore x=-1$