Correct option is (C) x = 2, y = 3
Given equations are
\(2^x+3^y=31\) __________(1)
and \(2^{x+2}-3^{y+1}=-65\)
\(\Rightarrow\) \(2^x.2^2-3^y.3=-65\)
\(\Rightarrow\) \(4.2^x-3.3^y=-65\) __________(2)
Take \(2^x=X\;\&\;3^y=Y\)
Then equations (1) & (2) converts into
X+Y = 31 __________(3)
and 4X - 3Y = -65 __________(4)
Put Y = 31 - X from equation (3) into equation (4), we obtain
4X - 3 (31 - X) = -65
\(\Rightarrow\) 4X - 93 + 3X = -65
\(\Rightarrow\) 7X = -65 + 93 = 28
\(\Rightarrow\) X = \(\frac{28}7\) = 4
\(\therefore\) Y = 31 - X
= 31 - 4 = 27
\(\Rightarrow2^x=4\;\&\;3^y=27\) \((\because X=2^x\;\&\;Y=3^y)\)
\(\Rightarrow2^x=2^2\;\&\;3^y=3^3\)
\(\Rightarrow\) x = 2 & y = 3
