Question

Solve the eqaution.
$\frac{5}{x - 1} + \frac{1}{y - 2} = 2$


and
$\frac{6}{x - 1} - \frac{3}{y - 2} = 1$

​

Answer 1

Step-by-step explanation:

First of all the method that we are going to use is grandly referred as substitution method.

Now, what we need to do is make assumptions with the denominator.

[Tip - We always make assumptions mostly with denominator in these type of questions]

Let $\tt{\dfrac{1}{x-a} = a}$ & $\tt{\dfrac{1}{y-2} = b}$

Now, let's see what happens next:-

5a + b = 2

5a = 2 - b

a = $\tt{\dfrac{2-b}{a}}$ --------------(1)

Second equation :-

6a - 3b = 1

Substitution of (1) here,

$\tt{6(\dfrac{2-b}{5}) - 3b = 5}$

[Take LCM as 5]

12 - 6b - 15b = 5

12 - 21b = 5

-21b = -7

b = $\tt{\dfrac{1}{3}}$

Don't forget that b is just an illusion, we need to bring the real values!

$\tt{\dfrac{1}{y-2} = b}$

$\tt{\dfrac{1}{y-2} = \dfrac{1}{3}}$

y - 2 = 3

y = 5 is the answer.

For x,

substitute the value of b in (1)

$\tt{ a = \dfrac{2 - \dfrac{1}{3}}{5}}$

$\tt{ a = \dfrac{\dfrac{6-1}{3}}{5}}$

$\tt{ a = \dfrac{\dfrac{5}{3}}{5}}$

$\tt{\dfrac{1}{5} = a}$

$\tt{\dfrac{1}{5} = \dfrac{1}{x-1}}$

x - 1 = 5

x = 6 is the answer.

Answer 2

Answer:

Solution :-

Let

5/x - p = p

and

1/y - 2 = q

So,

Now

5 × p + q = 2

5p + q = 2

5p = 2 - q

p = 2 - q/5

Now

6p - 3q = 1

6(2 - q/5) - 3q = 1

6 × 2 - q/5 - 3q = 1

12 - 6q/5 - 3b

12 - 6q - 15b/5 = 1

12 - 21q/5 = 1

12 - 21q = 5 × 1

12 - 21q = 5

12 - 5 = 21q

7 = 21q

7/21 = q

1/3 = q

1/y - 2 = 1/3

y - 2 = 1(3)

y - 2 = 3

y = 2 + 3

y = 5

Now

p = 2 - (1/3)/5

p = 2(3) - 1/3/5

p = 6-1/3/5

p = 5/3/5

p = 1/5

1/5 = 1/x - 1

x - 1 = 5(1)

x - 1 = 5

x = 1 + 5

x = 6