Solve: x+y=7xy and 2/x +3/y=17

Solve: x+y=7xy and 2/x +3/y=17

Elemination method

x+y=7xy —(i)

$\frac{2}{x} + \frac{3}{y} = 17$ —(ii)

From Equation (i) we get,

$\frac{x + y}{xy} = 7$

$\Rightarrow \frac{x}{xy} + \frac{y}{xy} = 7$

$\Rightarrow \frac{1}{y} + \frac{1}{x} = 7$ —(iii)

Multiplying Equation (iii) by 2 we get,

$\frac{2}{x} + \frac{2}{y} = 14$ —-(iv)

Now Subtracting Equation (iv) from equation (ii) we get,

$\left(\frac{2}{x} + \frac{3}{y}\right) – \left(\frac{2}{x} + \frac{2}{y}\right) = 17 – 14$

$\Rightarrow \frac{2}{x} + \frac{3}{y} – \frac{2}{x} – \frac{2}{y} = 3$

$\Rightarrow \frac{3}{y} – \frac{2}{y} = 3$

$\Rightarrow \frac{1}{y} = 3$

$\Rightarrow y = \frac{1}{3}$

Substituting the value of y in equation (iii) we get,

$\frac{1}{y} + \frac{1}{x} = 7$

$\Rightarrow \frac{1}{\left(\frac{1}{3}\right)} + \frac{1}{x} = 7$

$\Rightarrow 3 + \frac{1}{x} = 7$

$\Rightarrow \frac{1}{x} = 7 – 3$

$\Rightarrow \frac{1}{x} = 4$

$\Rightarrow {\mathrm{x}} = \frac{1}{4}$

Therefore Required Solutions are ${\mathrm{x}} = \frac{1}{4},{\mathrm{y}} = \frac{1}{3}$

Substitution Method

x+y=7xy —(i)

$\frac{2}{x} + \frac{3}{y} = 17$ —(ii)

From Equation (i) we get,

x=7xy-y

⟹ x = y(7x-1)

⟹${\mathrm{y}} = \frac{{\mathrm{x}}}{7{\mathrm{x}} – 1}$ —(iii)

Substituting the value of y in equation (ii) we get,

$\frac{2}{x} + \frac{3}{y} = 17$

$\Rightarrow \frac{2}{x} + \frac{3}{\left(\frac{x}{7x – 1}\right)} = 17$

$\Rightarrow \frac{2}{x} + \frac{3(7x – 1)}{x} = 17$

$\Rightarrow \frac{2}{x} + \frac{21x – 3}{x} = 17$

$\Rightarrow \frac{2 + 21x – 3}{x} = 17$

$\Rightarrow \frac{21x – 1}{x} = 17$

$\Rightarrow 21x – 1 = 17x$

$\Rightarrow 21x – 17x = 1$

$\Rightarrow 4x = 1$

$\Rightarrow x = \frac{1}{4}$

From equation (iii) we get,

${\mathrm{y}} = \frac{{\mathrm{x}}}{7{\mathrm{x}} – 1}$

$\Rightarrow {\mathrm{y}} = \frac{\frac{1}{4}}{\frac{7}{4} – 1}$

$\Rightarrow {\mathrm{y}} = \frac{\frac{1}{4}}{\frac{7 – 4}{4}}$

$\Rightarrow {\mathrm{y}} = \frac{\frac{1}{4}}{\frac{3}{4}}$

$\Rightarrow {\mathrm{y}} = \frac{1}{4} \div \frac{3}{4}$

$\Rightarrow {\mathrm{y}} = \frac{1}{4} \times \frac{4}{3}$

$\Rightarrow {\mathrm{y}} = \frac{1}{3}$

Therefore Required Solutions are ${\mathrm{x}} = \frac{1}{4},{\mathrm{y}} = \frac{1}{3}$