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If $5^{x}-3^{y}=13438$ and $5^{x-1}+3^{y+1}=9686$, then  $x+y$ equals _______

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 $:13$

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