Solve: 2^x+3^y=17,2^ x+2 -3^ y+1 =5 - Vidyadip

Question

Solve: $2^x+3^y=17,2^{x+2}-3^{y+1}=5$

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Answer

$2^x+3^y=17$...(i)
$2^{x+2}-3^{y+1}=5$
$\therefore \quad 2^x \cdot 2^2-3^y \cdot 3^1=5$
$\therefore \quad 4.2^x-3.3^y=5$ ...(ii)
Substituting $2^x=m$ and $3^y=n$, we get,
$m+n=17$...(iii)
$4 m-3 n=5$...(iv)
Multiplying eq. (iii) by 3,
3m + 3n = 51...(v)
Adding (iv) and (v) we get,

$\therefore \quad m=\frac{56}{7}$
$\therefore \quad m=8$
Substituting $m=8$ in (iii),
$8+n=17$
$\therefore \quad n=17-8$
$\therefore \quad n=9$
Resubstituting the value of m and n, we get,
$\begin{aligned} & m=2^x \\ \therefore & 8=2^x \\ \therefore & 2^3=2^x \\ \therefore & x=3\end{aligned}$
$\begin{array}{l}n=3^y \\ 9=3^y \\ 3^2=3^y \\ y=2\end{array}$
$\therefore$ $x=3$ and $y=2$ is the solution.

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