$${( \sqrt{3} )}^{x + y} = 9 \\ {( \sqrt{2} )}^{x – y} = 32$$ find the value of 2x + y.​

Question

$${( \sqrt{3} )}^{x + y} = 9 \\ {( \sqrt{2} )}^{x – y} = 32$$

find the value of 2x + y.​

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1 year 2021-09-03T15:08:25+00:00 1 Answers 5 views 0

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

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

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

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

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

$$\\$$

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

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

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

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

$$⇒ x + y = 4 \: \: \: \: (iii)$$

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

$${( \sqrt{2} )}^{x – y} = 32$$

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

$$⇒x – y = 10 \: \: \: \: (iv)$$

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

$$x + y + x – y = 4 + 10$$

$$⇒2x = 14$$

$$⇒x = 7$$

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

$$x – y – x – y = 10 – 4$$

$$⇒ – 2y = 6$$

$$⇒y = – 3$$

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

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

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

$$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: =14 – 3$$

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

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