Question

2/3x+2y+3/3x-2y=17/5 and 5/3x+2y+1/3x-2y=2​

Answer 1

Answer

$\sf \dashrightarrow \dfrac{2}{3x + 2y} + \dfrac{3}{3x - 2y} = \dfrac{17}{5} \: \: --- (i)$

$\sf \dashrightarrow \dfrac{5}{3x + 2y} + \dfrac{1}{3x - 2y} = 2 \: \: --- (ii)$

Let $\sf \dfrac{1}{3x + 2y}$ be u.

Let $\sf \dfrac{1}{3x - 2y}$ be v.

So, the equations become

$\sf \dashrightarrow 2u + 3v = \dfrac{17}{5}$

$\sf \dashrightarrow 5u + v = 2$

By first equation,

$\sf \dashrightarrow 2u + 3v = \dfrac{17}{5}$

$\sf \dashrightarrow 2u = \dfrac{17}{5} - 3v$

$\sf \dashrightarrow 2u = \dfrac{17 - 15v}{5}$

$\sf \dashrightarrow u = \dfrac{17 - 15v}{5} \times \dfrac{1}{2}$

$\sf \dashrightarrow u = \dfrac{17 - 15v}{10}$

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

$\sf \dashrightarrow 5u + v = 2$

$\sf \dashrightarrow 5 \bigg( \dfrac{17 - 15v}{10} \bigg) + v = 2$

$\sf \dashrightarrow \dfrac{85 - 75v}{10} + v = 2$

$\sf \dashrightarrow \dfrac{85 - 75v + 10v}{10} = 2$

$\sf \dashrightarrow \dfrac{85 - 65v}{10} = 2$

$\sf \dashrightarrow 85 - 65v = 2 \times 10$

$\sf \dashrightarrow 85 - 65v = 20$

$\sf \dashrightarrow -65v = 20 - 85$

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

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

$\sf \dashrightarrow v = 1$

Now, we can find the value of u by first equation.

$\sf \dashrightarrow 2u + 3v = \dfrac{17}{5}$

$\sf \dashrightarrow 2u + 3(1) = \dfrac{17}{5}$

$\sf \dashrightarrow 2u + 3 = \dfrac{17}{5}$

$\sf \dashrightarrow 2u = \dfrac{17}{5} - 3$

$\sf \dashrightarrow 2u = \dfrac{17 - 15}{5}$

$\sf \dashrightarrow 2u = \dfrac{2}{5}$

$\sf \dashrightarrow u = \dfrac{2}{5} \times \dfrac{1}{2}$

$\sf \dashrightarrow u = \dfrac{2}{10}$

$\sf \dashrightarrow u = \dfrac{1}{5}$

We know that,

$\sf \dashrightarrow \dfrac{1}{3x + 2y} = u$

$\sf \dashrightarrow \dfrac{1}{3x + 2y} = \dfrac{1}{5}$

$\sf \dashrightarrow 3x + 2y = 5 \: \: --- (iii)$

We also know that,

$\sf \dashrightarrow \dfrac{1}{3x - 2y} = v$

$\sf \dashrightarrow \dfrac{1}{3x - 2y} = 1$

$\sf \dashrightarrow 3x - 2y = 1 \: \: --- (iv)$

By third equation,

$\sf \dashrightarrow 3x + 2y = 5$

$\sf \dashrightarrow 3x = 5 - 2y$

$\sf \dashrightarrow x = \dfrac{5 - 2y}{3}$

Now, let's find the value of y by fourth equation.

$\sf \dashrightarrow 3x - 2y = 1$

$\sf \dashrightarrow 3 \bigg( \dfrac{5 - 2y}{3} \bigg) - 2y = 1$

$\sf \dashrightarrow \dfrac{15 - 6y}{3} - 2y = 1$

$\sf \dashrightarrow \dfrac{15 - 6y - 6y}{3} = 1$

$\sf \dashrightarrow \dfrac{15 - 12y}{3} = 1$

$\sf \dashrightarrow 15 - 12y = 3$

$\sf \dashrightarrow -12y = 3 - 15$

$\sf \dashrightarrow -12y = -12$

$\sf \dashrightarrow y = \dfrac{-12}{-12}$

$\sf \dashrightarrow y = 1$

Now, let's find the value of x by third equation.

$\sf \dashrightarrow 3x + 2y = 5$

$\sf \dashrightarrow 3x + 2(1) = 5$

$\sf \dashrightarrow 3x + 2 = 5$

$\sf \dashrightarrow 3x = 5 - 2$

$\sf \dashrightarrow 3x = 3$

$\sf \dashrightarrow x = \dfrac{3}{3}$

$\sf \dashrightarrow x = 1$

Hence, the values of x and y are 1 and 1 respectively.