Question

Solve the following equations
5/x - 1 - 1/y-2 = 2
6/x-1 - 3/y-2 = 1​

Answer 1

Answer

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

$\sf \dashrightarrow \dfrac{6}{x - 1} - \dfrac{3}{y - 2} = 1 \: \: --- (ii)$

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

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

So, the equations become

$\sf \dashrightarrow 5u - 1v = 2$

$\sf \dashrightarrow 6u - 3v = 1$

By first equation,

$\sf \dashrightarrow 5u - 1v = 2$

$\sf \dashrightarrow 5u = 2 + 1v$

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

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

$\sf \dashrightarrow 6u - 3v = 1$

$\sf \dashrightarrow 6 \bigg( \dfrac{2 + 1v}{5} \bigg) - 3v = 1$

$\sf \dashrightarrow \dfrac{12 + 6v}{5} - 3v = 1$

$\sf \dashrightarrow \dfrac{12 + 6v - 15v}{5} = 1$

$\sf \dashrightarrow \dfrac{12 - 9v}{5} = 1$

$\sf \dashrightarrow 12 - 9v = 5$

$\sf \dashrightarrow -9v = 5 - 12$

$\sf \dashrightarrow -9v = -7$

$\sf \dashrightarrow v = \dfrac{-7}{-9}$

$\sf \dashrightarrow v = \dfrac{7}{9}$

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

$\sf \dashrightarrow 5u - 1v = 2$

$\sf \dashrightarrow 5u - 1 \bigg( \dfrac{7}{9} \bigg) = 2$

$\sf \dashrightarrow 5u - \dfrac{7}{9} = 2$

$\sf \dashrightarrow \dfrac{45u - 7}{9} = 2$

$\sf \dashrightarrow 45u - 7 = 9 \times 2$

$\sf \dashrightarrow 45u - 7 = 18$

$\sf \dashrightarrow 45u = 18 + 7$

$\sf \dashrightarrow 45u = 25$

$\sf \dashrightarrow u = \dfrac{25}{45}$

$\sf \dashrightarrow u = \dfrac{5}{9}$

We know that,

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

$\sf \dashrightarrow \dfrac{1}{x - 1} = \dfrac{5}{9}$

$\sf \dashrightarrow 9 = 5(x - 1)$

$\sf \dashrightarrow 5x - 5 = 9$

$\sf \dashrightarrow 5x = 9 + 5$

$\sf \dashrightarrow 5x = 14$

$\sf \dashrightarrow x = \dfrac{14}{5}$

We also know that,

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

$\sf \dashrightarrow \dfrac{1}{y - 2} = \dfrac{7}{9}$

$\sf \dashrightarrow 9 = 7(y - 2)$

$\sf \dashrightarrow 7y - 14 = 9$

$\sf \dashrightarrow 7y = 9 + 14$

$\sf \dashrightarrow 7y = 23$

$\sf \dashrightarrow y = \dfrac{23}{7}$

Hence, the values of x and y are 14/5 and 23/7 respectively.