Question

Csqrt%7B2%7D%20%29%7D%5E%7Bx%20-%20y%7D%20%20%3D%2032" id="TexFormula1" title=" {( \sqrt{3} )}^{x + y} = 9 \\ {( \sqrt{2} )}^{x - y} = 32" alt=" {( \sqrt{3} )}^{x + y} = 9 \\ {( \sqrt{2} )}^{x - y} = 32" align="absmiddle" class="latex-formula">
find the value of 2x + y.​

Answer 1

$\bf \purple{ \underline{Given :-}}$

$• \: {( \sqrt{3} )}^{x + y} = 9\: \: \: \: (i)$

$• \: {( \sqrt{2} )}^{x - y} = 32 \: \: \: \: (ii)$

$\bf \purple{ \underline{To \: Find :- }}$

$• {\sf{The \: value \: of}} \: \: 2x+ y.$

$\huge\bf \purple{ \underline{Solution :- }}$

$\sf{From \: equation \: (i), }$

${( \sqrt{3} )}^{x + y} = 9$

$⇒ {( \sqrt{3} )}^{x + y} = ({ \sqrt{3} })^{4}$

$⇒ x + y = 4 \: \: \: \: (iii)$

$\sf{From \: the \: equation \: (ii)}$

${( \sqrt{2} )}^{x - y} = 32$

$⇒( { \sqrt{2} })^{x - y} =( { \sqrt{2} })^{10}$

$⇒x - y = 10 \: \: \: \: (iv)$

$\sf{We \: have \: to \: add \: equation \: (iii) \: and \: equation \: (iv)}$

$x + y + x - y = 4 + 10$

$⇒2x = 14$

$⇒x = 7$

$\sf{Subtracting \: equation \: (iii) \: from \: equation \: (ii), }$

$x - y - x - y = 10 - 4$

$⇒ - 2y = 6$

$⇒y = - 3$

$\bf\therefore x = 7 \: \: and \: \: y = - 3$

${ \sf{Th e \: value \: of }}= 2x +y$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: =2 \times 7 + ( - 3)$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: =14 - 3$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: =11$

$\bf \red{Hence, \: the \: value \: of \: 2x + y \: is \: 11. }$