Given that,
- $5^{x} -3^{y}=13438 \quad \longrightarrow (1)$
- $5^{x-1}-3^{y+1}=9686 \quad \longrightarrow (2)$
$5^{x} \implies x=1,2,3,4,5,6,7, \dots$
- $5^{1}=5$
- $5^{2}=25$
- $5^{3}=125$
- $5^{4}=625$
- $5^{5}=3125$
- $5^{6}=15625$
- $5^{7}=78125$
$3^{y} \implies y=1,2,3,4,5,6,7,8, \dots$
- $3^{1}=3$
- $3^{2}=9$
- $3^{3}=27$
- $3^{4}=81$
- $3^{5}=243$
- $3^{6}=729$
- $3^{7}=2187$
- $3^{8}=6561$
Put the values of $x$ and $y$ in equation $(1)$ and $(2).$ And check which values of $x$ and $y$ are satisfies.
Now, lets take $x = 6,y = 7,$ and check
- $5^{x}- 3^{y}=13438 \quad \longrightarrow (1)$
- $ 5^{6}-3^{7}=13438$
- $15625-2187=13438$
- $13438 =13438\quad $ (The first equation is satisfied)
Similarly, we can check for equation $(2).$
- $5^{x-1}-3^{y+1}=9686 \quad \longrightarrow (2)$
- $5^{6-1}-3^{7+1}=9686$
- $5^{5}-3^{8}=9686$
- $3125+6561=9686$
- $9686=9686 \quad $ (The second equation is satisfied)
We get, the value of $x=6,y=7$
$\therefore x+y=6+7=13$
$\textbf{Second Method:}$
Given that,
- $5^{x}-3^{y}=13438 \quad \longrightarrow (1)$
- $5^{x-1}+3^{y+1}=9686 \quad \longrightarrow (2)$
We can rewrite the equation $(2).$
- $5^{x} \ast 5^{-1}+ 3^{y}\ast 3^{1}=9686$
- $\frac{5^{x}}{5}+ 3^{y} \ast 3=9686 \quad \longrightarrow (3)$
We can multiply $`3\text{’}$ both sides in equation $(1).$
- $5^{x} \ast 3 – 3^{y} \ast 3=13438.3$
- $5^{x} \ast 3 – 3^{y}\ast 3=40314 \quad \longrightarrow (4) $
By adding the equation $(3)$ and $(4)$, we get
$\Rightarrow \frac{5^{x}}{5} +3^{y}\ast 3+5^{x}\ast 3-3^{y}\ast 3=9686+40314$
$ \Rightarrow 5^{x}\ast 5^{-1}+5^{x}\ast 3^{1}=50000$
$ \Rightarrow 5^{x} \left(5^{-1}+3^{1} \right)=50000$
$ \Rightarrow 5^{x} \left (\frac{1}{5}+3 \right)=50000$
$ \Rightarrow 5^{x} \left( \frac{16}{5} \right)=50000$
$ \Rightarrow 5^{x}= \frac{250000}{16}$
$ \Rightarrow 5^{x}=15625$
$ \Rightarrow5^{x} = 5^{6}$
$\implies \boxed{x=6}$
We can put the value of $`x\text{’}$ in equation $(1)$, we get
$\Rightarrow 5^{x}-3^{y}=13438$
$ \Rightarrow 5^{6}-3^{y}=13438$
$ \Rightarrow 15625-3^{y}=13438$
$ \Rightarrow -3^{y}=13438-15625$
$ \Rightarrow -3^{y} =-2187$
$\Rightarrow 3^{y}=2187$
$ \Rightarrow 3^{y}=3^{7}$
$\Rightarrow \boxed{y=7} $
$\therefore$ The value of $x+y=6+7=13$
Correct Answer $:13$