Solve the equation: $\;$ $7 \times 3^{x+1} - 5^{x+2} = 3^{x+4} - 5^{x+3}$
Given equation: $\;\;$ $7 \times 3^{x+1} - 5^{x+2} = 3^{x+4} - 5^{x+3}$
i.e. $\;$ $7 \times 3^{x+1} - 3^{x+1} \times 3^3 = 5^{x+2} - 5^{x+2} \times 5^1$
i.e. $\;$ $3^{x+1} \left(7 - 27\right) = 5^{x+2} \left(1 - 5\right)$
i.e. $\;$ $3^{x+1} \times \left(-20\right) = 5^{x+1} \times 5 \times \left(-4\right)$
i.e. $\;$ $\left(\dfrac{3}{5}\right)^{x+1} = 1$
i.e. $\;$ $\left(\dfrac{3}{5}\right)^{x+1} = \left(\dfrac{3}{5}\right)^0$
$\implies$ $x + 1 = 0$ $\implies$ $x = -1$
$\therefore \;$ The solution to the given equation is $\;\;$ $x = -1$