Question

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Answer 1

Answer

$\sf \dashrightarrow \dfrac{3}{2x - y} + \dfrac{8}{x + 2y} = 3 \: \: --- (i)$

$\sf \dashrightarrow \dfrac{12}{x + 2y} - \dfrac{6}{2x - y} = 1 \: \: --- (ii)$

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

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

So, the equations become

$\sf \dashrightarrow 3u + 8v = 3$

$\sf \dashrightarrow 12v - 6u = 1$

By first equation,

$\sf \dashrightarrow 3u + 8v = 3$

$\sf \dashrightarrow 3u = 3 - 8v$

$\sf \dashrightarrow u = \dfrac{3 - 8v}{3}$

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

$\sf \dashrightarrow 12v - 6u = 1$

$\sf \dashrightarrow 12v - 6 \bigg( \dfrac{3 - 8v}{3} \bigg) = 1$

$\sf \dashrightarrow 12v - \dfrac{18 - 48v}{3} = 1$

$\sf \dashrightarrow \dfrac{36v - 18 - 48v}{3} = 1$

$\sf \dashrightarrow \dfrac{-12v - 18}{3} = 1$

$\sf \dashrightarrow -12v - 18 = 3$

$\sf \dashrightarrow -12v = 3 + 18$

$\sf \dashrightarrow -12v = 21$

$\sf \dashrightarrow v = \dfrac{21}{-12}$

$\sf \dashrightarrow v = \dfrac{-7}{4}$

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

$\sf \dashrightarrow 3u + 8v = 3$

$\sf \dashrightarrow 3u + 8 \bigg( \dfrac{-7}{4} \bigg) = 3$

$\sf \dashrightarrow 3u + \dfrac{-56}{4} = 3$

$\sf \dashrightarrow 3u + (-14) = 3$

$\sf \dashrightarrow 3u - 14 = 3$

$\sf \dashrightarrow 3u = 3 + 14$

$\sf \dashrightarrow 3u = 17$

$\sf \dashrightarrow u = \dfrac{17}{3}$

We know that,

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

$\sf \dashrightarrow \dfrac{1}{2x - y} = \dfrac{17}{3}$

$\sf \dashrightarrow 3 = 17(2x - y)$

$\sf \dashrightarrow 34x - 17y = 3 \: \: --- (iii)$

We also know that,

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

$\sf \dashrightarrow \dfrac{1}{x + 2y} = \dfrac{-7}{4}$

$\sf \dashrightarrow 4 = -7(x + 2y)$

$\sf \dashrightarrow -7x -14y = 4 \: \: --- (iv)$

Now, by third equation,

$\sf \dashrightarrow 34x - 17y = 3$

$\sf \dashrightarrow 34x = 3 + 17y$

$\sf \dashrightarrow x = \dfrac{3 + 17y}{34}$

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

$\sf \dashrightarrow -7x - 14y = 4$

$\sf \dashrightarrow -7 \bigg( \dfrac{3 + 17y}{34} \bigg) - 14y = 4$

$\sf \dashrightarrow \dfrac{-21 - 119y}{34} - 14y = 4$

$\sf \dashrightarrow \dfrac{-21 - 119y - 476y}{34} = 4$

$\sf \dashrightarrow \dfrac{-21 - 595y}{34} = 4$

$\sf \dashrightarrow -21 - 595y = 4 \times 34$

$\sf \dashrightarrow -21 - 595y = 136$

$\sf \dashrightarrow -595y = 136 + 21$

$\sf \dashrightarrow -595y = 157$

$\sf \dashrightarrow y = \dfrac{157}{-595}$

$\sf \dashrightarrow y = \dfrac{-157}{595}$

Now, we can find the value of x by third equation.

$\sf \dashrightarrow 34x - 17y = 3$

$\sf \dashrightarrow 34x - 17 \bigg( \dfrac{-157}{595} \bigg) = 3$

$\sf \dashrightarrow 34x - \dfrac{-2669}{595} = 3$

$\sf \dashrightarrow \dfrac{20230x + 2669}{595} = 3$

$\sf \dashrightarrow 20230x + 2669 = 595 \times 3$

$\sf \dashrightarrow 20230x + 2669 = 1785$

$\sf \dashrightarrow 20230x = 1785 - 2669$

$\sf \dashrightarrow 20230x = -884$

$\sf \dashrightarrow x = \dfrac{-884}{20230}$

$\sf \dashrightarrow x = \dfrac{-442}{10115}$

Hence, the values of x and y are -442/10115 and -157/595 respectively.

Answer 2

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