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The solutions of x and y for which the given equations hold are and respectively.

$6 x + 3 y = 6 xy$ and $2 x + 4 y = 5 xy$

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Detailed Solution

Given:

6 x + 3 y = 6 xy

and

2 x + 4 y = 5 x

(\frac{6 x + 3 y}{xy}) = 6

and

(\frac{2 x + 4 y}{xy}) = 5

(\frac{6 x}{xy} + \frac{3 y}{xy}) = 6

and

(\frac{2 x}{xy} + \frac{4 y}{xy}) = 5

(\frac{6}{y} + \frac{3}{x}) = 6

and

(\frac{2}{y} + \frac{4}{x}) = 5

Put

\frac{1}{x} = a

and

\frac{1}{y} = b

, we get

6 b + 3 a = 6

and

2 b + 4 a = 5

Multiply

2 b + 4 a = 5

with 3 and subtract

6 b + 3 a = 6

from it,
Thus, we get

6 b + 12 a - (6 b + 3 a) = 15 - 6

9a = 9
a = 1
Therefore, 6b + 3(1) = 6
6b = 3
b =

\frac{1}{2}

Thus,

\frac{1}{x} = 1

and

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

which implies, x = 1 and y = 2
Hence, the value of x is 1 and y is 2.

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