Question

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.

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

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

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

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

$\sf \dashrightarrow \dfrac{12 - 21v}{5} = 1$

$\sf \dashrightarrow 12 - 21v = 5$

$\sf \dashrightarrow -21v = 5 - 12$

$\sf \dashrightarrow -21v = -7$

$\sf \dashrightarrow v = \dfrac{-7}{-21}$

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

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

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

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

$\sf \dashrightarrow \dfrac{15u + 1}{3} = 2$

$\sf \dashrightarrow 15u + 1 = 3 \times 2$

$\sf \dashrightarrow 15u + 1 = 6$

$\sf \dashrightarrow 15u = 6 - 1$

$\sf \dashrightarrow 15u = 5$

$\sf \dashrightarrow u = \dfrac{5}{15}$

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

We know that,

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

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

$\sf \dashrightarrow x - 1 = 3$

$\sf \dashrightarrow x = 3 + 1$

$\sf \dashrightarrow x = 4$

We also know that,

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

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

$\sf \dashrightarrow y - 2 = 3$

$\sf \dashrightarrow y = 3 + 2$

$\sf \dashrightarrow y = 5$

Hence, the values of x and y are 4 and 5 respectively.

Answer 2

Answer:

x = 4 and y = 5

Step-by-step explanation:

Given: [5/(x–1)] + [1/(y– 2)] = 2 , [6/(x–1)] – [3/(y–2)] = 1

Let 1/(x-1) = u and 1/(y-2) = v

Then we get equations as:

5u + v = 2 -------(1)

6u - 3v = 1 ------(2)

(1) * 3 ==> 15u + 3v = 6

(2)       ==> 6u    - 3v = 1

Adding==> 21u = 7

u = 7/21 = 1/3

Substuting u in equation 2, we get:

6 (1/3) - 3v = 1

-3v = 1-2

v = -1/-3 = 1/3

1/x-1 = 1/3, so x-1 = 3, x = 4.

1/y-2 = 1/3, so y-2 = 3, y = 5.

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