Question

number of positive integral solutions o the equation 1/x+2/y=1/4 A)4 B)6 C)8 D)10

Answer 1

Given :  1/x+2/y=1/4  

To find : number of positive integral solutions of the equation

Solution:

1/x+2/y=1/4

=> 4y + 8x = xy

=> 8x = xy - 4y

=> 8x = y(x - 4)

=> y  =  8x/(x - 4)

=> x >  4    or  or x = 4y/(y - 8)  => y  > 8  

=> x = 5 , y = 40

   x = 6 ,  y = 24

   x = 8  , y = 16

  x = 12  , y  = 12

  x = 20 ,  y = 10

  x = 36  , y  = 9

6 positive integral Solutions

6  number of positive integral solutions of the equation

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Answer 2

$\frac{1}{x} + \frac{2}{y} = \frac{1}{4}$

$x,y \geq 1$

$\implies \frac{1}{x} = \frac{1}{4} - \frac{2}{y}$

$\implies \frac{1}{x} = \frac{y - 8}{4y}$

$\implies x = \frac{4y}{y - 8}$

$\implies x = \frac{4(y - 8) + 32}{y - 8}$

$\implies x = 4 + \frac{32}{y - 8}$

Now, (y–8) must be a factor of 32.

Thus, Finding the Number of Factors will provide the number of solutions to the given equation.

$32 = {2}^{5}$

Therefore, the number of factors of 32 are:

$\boxed{n(x : x| {2}^{5} ) = ^{5}\sf{C}_{1} + \: ^{5}C_{0} = 5 + 1 = 6}$

Therefore, There are 6 positive integral solutions.

They can be obtained by equating (y-8) to each factor of 32 and then obtaining value of x

1. y-8=1

y=9

x=4+32 =36

.

2. y-8=2

y=10

x=4+16=20

.

3. y-8=4

y=12

x=4+8=12

.

4. y-8=8

y=16

x=8

.

5. y-8=16

y=24

x=6

.

6. y-8=32

y=40

x=5

Therefore,

(x,y)∈ {(36,9),(20,10),(12,12),(8,16),(6,24),(5,40)}