Question

$x = \frac{2}{2 - \frac{1}{5} - \frac{1}{2}} - 1 \\ y = \frac{1}{3 - \frac{1}{3} - \frac{1}{4} + 1 }$
Find:
$\frac{x}{3} - \frac{y}{2}$
​

Answer 1

Given :

• $\sf x = \dfrac{2}{2 - \dfrac{1}{5} - \dfrac{1}{2}} - 1$

• $\sf y = \dfrac{1}{3 - \dfrac{1}{3} - \dfrac{1}{4}} + 1$

To Find :

• $\sf \dfrac{x}{3} - \dfrac{y}{2}$

Solution :

We have :

• $\sf x = \dfrac{2}{2 - \dfrac{1}{5} - \dfrac{1}{2}} - 1 \: -(1)$

• $\sf y = \dfrac{1}{3 - \dfrac{1}{3} - \dfrac{1}{4}} + 1 \: -(2)$

Solving equation (1) :

$\sf : \implies x = \dfrac{2}{2 - \dfrac{1}{5} - \dfrac{1}{2}} - 1$

$\sf : \implies x = \dfrac{2}{\dfrac{2\times 10}{10} - \dfrac{1 \times 2}{5 \times 2} - \dfrac{1 \times 5}{2 \times 5}} - 1$

$\sf : \implies x = \dfrac{2}{\dfrac{20}{10} - \dfrac{2}{10} - \dfrac{5}{10}} - 1$

$\sf : \implies x = \dfrac{2}{\dfrac{20- 2 - 5}{10}} - 1$

$\sf : \implies x = \dfrac{2}{\dfrac{20 - 7}{10}} - 1$

$\sf : \implies x = \dfrac{2}{\dfrac{13}{10}} - 1$

$\sf : \implies x = \dfrac{2}{13} \times 10 - 1$

$\sf : \implies x = \dfrac{20}{13} - 1$

$\sf : \implies x = \dfrac{20}{13} - \dfrac{13}{13}$

$\sf : \implies x = \dfrac{20 - 13}{13}$

$\sf : \implies x = \dfrac{7}{13}$

$\Large \underline{\boxed{\bf{x = \dfrac{7}{13}}}}$

Now, solving equation (2) :

$\sf : \implies y = \dfrac{1}{3 - \dfrac{1}{3} - \dfrac{1}{4}} + 1$

$\sf : \implies y = \dfrac{1}{\dfrac{3 \times 12}{12} - \dfrac{1 \times 4}{3 \times 4} - \dfrac{1 \times 3}{4 \times 3}} + 1$

$\sf : \implies y = \dfrac{1}{\dfrac{36}{12} - \dfrac{4}{12} - \dfrac{3}{12}} + 1$

$\sf : \implies y = \dfrac{1}{\dfrac{36 - 4 - 3}{12}} + 1$

$\sf : \implies y = \dfrac{1}{\dfrac{36 - 7}{12}} + 1$

$\sf : \implies y = \dfrac{1}{\dfrac{29}{12}} + 1$

$\sf : \implies y = \dfrac{1}{29} \times 12 + 1$

$\sf : \implies y = \dfrac{1}{29} \times 12 + 1$

$\sf : \implies y = \dfrac{12}{29} + 1$

$\sf : \implies y = \dfrac{12}{29} + \dfrac{29}{29}$

$\sf : \implies y = \dfrac{12 + 29}{29}$

$\sf : \implies y = \dfrac{41}{29}$

$\Large \underline{\boxed{\bf{y = \dfrac{41}{29}}}}$

$\underline{\sf Now, \: let's \: find \: \dfrac{x}{3} - \dfrac{y}{2}} :$

By putting values of x and y :

$\sf : \implies \dfrac{\dfrac{7}{13}}{3} - \dfrac{\dfrac{41}{29}}{2}$

$\sf : \implies \dfrac{7}{13} \times 3 - \dfrac{41}{29} \times 2$

$\sf : \implies \dfrac{21}{13} - \dfrac{82}{29}$

$\sf : \implies \dfrac{21 \times 29}{13 \times 29} - \dfrac{82 \times 13}{29 \times 13}$

$\sf : \implies \dfrac{609}{377} - \dfrac{1066}{377}$

$\sf : \implies \dfrac{609 - 1066}{377}$

$\sf : \implies \dfrac{-457}{377}$

Hence, answer is $\bf \dfrac{-457}{377}$

Answer 2

Answer:

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