Question

Annyeong mates ❤

Kindly,Can someone please do me the two sums ?
m
i) 14/x+y + 3/x-y = 5

21/x+y - 1/x-y = 2

ii) x+a/a = y+b/b

ax- by = a²- b²

Kindly please do me the sums .... I'll be really grateful .... ❤❤​

Answer 1

$\bold\red\star$$\mathrm\green{Question:-}$

$1. \: \: \frac{14}{x + y} + \frac{3}{x - y} = 5 \: \: \\ sum \\ \frac{21}{x + y} - \frac{ 1}{x - y} = 2$

$\red\star$$\mathrm\green{Solution:-}$

$\mathrm{Here,}$

$\frac{14}{x + y} + \frac{3}{x - y} = 5....... \fbox \blue{equation 1}$

$\frac{21}{x + y} - \frac{1}{x - y} = 2........ \fbox \blue{equation2}$

$\mathrm{Let,\frac{1}{x+y}=U\:and \:\frac{1}{x-y}\:put}$

$\mathrm{these\:values\:in\:eqn\:1\:and\:eqn\:2}$

$14u + 3v = 5........ \fbox \blue{ equation \: 3}$

$21u - v = 2............ \fbox \blue{equation \: 4}$

$\mathrm{from\: equation\:3\:we\:get,}$

$14u + 3v = 5$

$➟ \: 14u = 5 - 3v$

$➟ \: u = \frac{5 - 3v}{14} .......... \fbox \blue{equation \: 5}$

$\mathrm{put\:the\: value\:of\:u\:in\: equation\:2}$

$21u - v = 2$

$➟ \: 21 \times ( \frac{5 - 3v}{14} ) - v = 2$

$➟ \: 105 - \frac{63v}{14} - v = 2$

$➟ \: 105 - 63v - 14v = 14 \times 2$

$➟ \: ( - 77v) = 28 - 105$

$➟ \: ( - 77v) = ( - 77)$

$➟ \: v = \frac{ - 77}{ - 77 }$

$➟ \fbox \pink{\: v = 1}$

$\mathrm{put\:the\: value\:of\:v\:in\: equation\:3}$

$u = \frac{5 - 3v}{14}$

$⇒ \: u = 5 - 3 \times \frac{1}{14}$

$⇒ \: \frac{5 - 3}{14}$

$⇒ \: \frac{2}{14}$

$⇒ \: \frac{1}{7}$

$\mathrm{Therefore,}$

$\frac{1}{(x + y)} = u$

$⇒ \frac{1}{x + y} = \frac{1}{7}$

$⇒ \: (x + y) = 7..... (1)$

$\mathrm{and,}$

$\frac{1}{x - y} = v$

$➟ \: \frac{1}{x - y} = 1$

$➟ \: x - y = 1......(2)$

$\mathrm{from\: equation\:(1)\:we\:get,}$

$\: \: \: \: \: \: \: \: \: \: \: \: x + y = 7$

$⇒ \: x = 7 - y.......(3)$

$\mathrm{put\:the\: value\:of\:x\:in\: equation\:1}$

$\: \: \: \: \: x - y = 1$

$⇒ \: 7 - y - y = 1$

$⇒ \: - 2y = \frac{1}{7}$

$⇒ \: - 2y = - 6$

$⇒ \fbox\red{y = 3}$

$\mathrm{put\:the\: value\:of\:y\:in\: equation\:3}$

$\fbox\red{x = 4}$

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