Question

Solve for x and y

1/2x - 1/y = 1

1/x + 1/2y = 8

​

Answer 1

Answer:

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Step-by-step explanation:

The value of x and y is \frac{1}{6}\ {and}\ \frac{1}{4}

6

1

and

4

1

.

To find:

Solve the Equation: \frac{1}{2 x}-\frac{1}{y}=-1 \text { and } \frac{1}{x}+\frac{1}{2 y}=8=?

2x

1

−

y

1

=−1 and

x

1

+

2y

1

=8=?

Solution:

Given: \frac{1}{2 x}-\frac{1}{y}=-1 \text { and } \frac{1}{x}+\frac{1}{2 y}=8

2x

1

−

y

1

=−1 and

x

1

+

2y

1

=8

\frac{1}{2 x}-\frac{1}{y}=-1

2x

1

−

y

1

=−1

\frac{y-2 x}{2 x y}=-1

2xy

y−2x

=−1

y-2 x=-2 x yy−2x=−2xy

2 \mathrm{x}-\mathrm{y}=2 \mathrm{xy} \ldots \ldots \ldots (1)2x−y=2xy………(1)

\frac{1}{x}+\frac{1}{2 y}=8

x

1

+

2y

1

=8

\frac{2 y+x}{2 x y}=8

2xy

2y+x

=8

2 y+x=16 x y2y+x=16xy

x+2 y=16 x y \dots \ldots \ldots \ldots(2)x+2y=16xy…………(2)

Multiplying equation (1) by 2 and then adding it to equation (2), we get

5 \mathrm{x}=20 \mathrm{xy}5x=20xy

y=\frac{1}{4}y=

4

1

Replacing the value of y=\frac{1}{4}y=

4

1

in equation (1),

2 x-y=2 x y2x−y=2xy

2 x-\frac{1}{4}=2 x y2x−

4

1

=2xy

8 x-1=\frac{8 x y}{4}8x−1=

4

8xy

6 \mathrm{x}=16x=1

x=\frac{1}{6}x=

6

1

Answer 2

Step-by-step explanation:

Given equations:-

$\rm\dfrac{1}{2x} - \dfrac{1}{y} = - 1$

$\rm\dfrac{1}{x} + \dfrac{1}{2y} = 8$

Let 1/x = a (and) 1/y = b

$\rm\dashrightarrow \dfrac{1}{2x} - \dfrac{1}{y} = - 1$

$\rm\dashrightarrow\dfrac{a}{2} - b= - 1$

$\rm \dashrightarrow a - 2b = -2.....(i)$

$\rm\dashrightarrow\dfrac{1}{x} + \dfrac{1}{2y} = 8$

$\rm \dashrightarrow a + \dfrac{b}{2} = 8$

$\rm \dashrightarrow2a + b = 16......(ii)$

Multiply equation (i) by 2

→ 2( a - 2b ) = -2

→ 2a - 4b = -4......(iii)

Subtracting equation (ii) from (iii)

→ 2a - 4b - (2a + b) = -4 - 16

→ 2a - 2a - 4b - b = -20

→ -5b = -20

→ b = 4

Substituting b = 4 in equation (i)

→ a - 2b = -2

→ a - 2(4) = -2

→ a = -2 + 8 = 6

→ x = 1/a = 1/6

→ y = 1/b = 1/4

$\boxed{ \rm \leadsto x = \dfrac{1}{6} \: , \: \: \: y = \dfrac{1}{4} }$