Question

If 2/x+3/y=13 and 5/x-4/y = -2, then x + y equals (a)⅙ (b)-1/6 (c)⅚ (d)-5/6​

Answer 1

Answer

$\sf \dashrightarrow \dfrac{2}{x} + \dfrac{3}{y} = 13 \: \: --- (i)$

$\sf \dashrightarrow \dfrac{5}{x} - \dfrac{4}{y} = -2 \: \: --- (ii)$

Let $\sf \dfrac{1}{x}$ be u.

Let $\sf \dfrac{1}{y}$ be v.

So, the equations become

$\sf \dashrightarrow 2u + 3v = 13$

$\sf \dashrightarrow 5u - 4v = -2$

By first equation,

$\sf \dashrightarrow 2u + 3v = 13$

$\sf \dashrightarrow 2u = 13 - 3v$

$\sf \dashrightarrow u = \dfrac{13 - 3v}{2}$

Now, let's find the value of v by second equation.

$\sf \dashrightarrow 5u - 4v = -2$

$\sf \dashrightarrow 5 \bigg( \dfrac{13 - 3v}{2} \bigg) - 4v = -2$

$\sf \dashrightarrow \dfrac{65 - 15v}{2} - 4v = -2$

$\sf \dashrightarrow \dfrac{65 - 15v - 8v}{2} = -2$

$\sf \dashrightarrow \dfrac{65 - 23v}{2} = -2$

$\sf \dashrightarrow 65 - 23v = -2 \times 2$

$\sf \dashrightarrow 65 - 23v = -4$

$\sf \dashrightarrow -23v = -4 - 65$

$\sf \dashrightarrow -23v = -69$

$\sf \dashrightarrow v = \dfrac{-69}{-23}$

$\sf \dashrightarrow v = 3$

Now, let's find the value of u by first equation.

$\sf \dashrightarrow 2u + 3v = 13$

$\sf \dashrightarrow 2u + 3(3) = 13$

$\sf \dashrightarrow 2u + 9 = 13$

$\sf \dashrightarrow 2u = 13 - 9$

$\sf \dashrightarrow 2u = 4$

$\sf \dashrightarrow u = \dfrac{4}{2}$

$\sf \dashrightarrow u = 2$

We know that,

$\sf \dashrightarrow \dfrac{1}{x} = u$

$\sf \dashrightarrow \dfrac{1}{x} = 2$

$\sf \dashrightarrow 2x = 1$

$\sf \dashrightarrow x = \dfrac{1}{2}$

We also know that,

$\sf \dashrightarrow \dfrac{1}{y} = v$

$\sf \dashrightarrow \dfrac{1}{y} = 3$

$\sf \dashrightarrow 3y = 1$

$\sf \dashrightarrow y = \dfrac{1}{3}$

Now, let's find the answer if this question.

$\sf \dashrightarrow x + y$

$\sf \dashrightarrow \dfrac{1}{2} + \dfrac{1}{3}$

$\sf \dashrightarrow \dfrac{3 + 2}{6}$

$\sf \dashrightarrow \dfrac{5}{6}$

Hence, the values of x+y is 5/6.