Question

By the question we can easily come across y=2. but how can we mathematically derive it ?

Answer 1

${2}^{x} + {3}^{y} = 17 ...(1)\\ {2}^{x + 2} - {3}^{y + 1} = 5 .....(2)\\ 2( {2}^{x} ) - 3( {3}^{y} ) = 5 ...(3)\\ \\ we \: have \: \\ {2}^{x} + {3}^{y} = 17 \\ 2( {2}^{x} ) - 3( {3}^{y} ) = 5 \\ \\ multiply \: (1) \: by \: 3 \: and \: then \: add \: both \: eqns \: \\ 3( {2}^{x} ) + 3( {3}^{y} ) = 3(17) \\ 2( {2}^{x} ) - 3( {3}^{y} ) = 5 \\ add \\ 5( {2}^{x} ) = 56 \\ {2}^{x} = \frac{56}{5} \\ put \: in \: (1) \\ \frac{56}{5} + {3}^{y} = 17 \\ {3}^{y} = 17 - \frac{56}{5} \\ {3}^{y} = \frac{85 - 56}{5} \\ from \: here \: u \: will \: get \: y \: \: \: \: \: \: \: \: \: \\ p \\ check \: mistake \: btw \: procedure \: will \: remain \: same$

Answer 2

let 2^x = a and 3^y = b
now we can write the equations as
a + b = 17 (1)
4a - 3b = 5 (2)
on simplying the above equations we'll get
a = 8 and b = 9
now since
2^x = a
so 2^x = 8
2^x = 2^3
so x= 3
similarly,
3^y = 3^2
so y = 2
Hope this helps!!!