Solve:8x-3y=5xy and 6x-5y=-2xy

Solve:8x-3y=5xy and 6x-5y=-2xy

Elemination Method

8x-3y=5xy —(i)

6x-5y=-2xy —(ii)

From Equation (i) we get,

$\frac{8{\mathrm{x}} – 3{\mathrm{y}}}{{\mathrm{xy}}} = 5$

$\Rightarrow \frac{8{\mathrm{x}}}{{\mathrm{xy}}} – \frac{3{\mathrm{y}}}{{\mathrm{xy}}} = 5$

$\Rightarrow \frac{8}{{\mathrm{y}}} – \frac{3}{{\mathrm{x}}} = 5$ —(iii)

From Equation (ii) we get,

$\frac{6{\mathrm{x}} – 5{\mathrm{y}}}{{\mathrm{xy}}} = – 2$

$\Rightarrow \frac{6{\mathrm{x}}}{{\mathrm{xy}}} – \frac{5{\mathrm{y}}}{{\mathrm{xy}}} = – 2$

$\Rightarrow \frac{6}{{\mathrm{y}}} – \frac{5}{{\mathrm{x}}} = – 2$—(iv)

Now Multiplying Equation (iii) by 5 and Equation (iv) by 3 we get,

$\frac{40}{{\mathrm{y}}} – \frac{15}{{\mathrm{x}}} = 25$ —-(v)

and $\frac{18}{{\mathrm{y}}} – \frac{15}{{\mathrm{x}}} = – 6$ —-(vi)

Now Subtracting Equation (vi) from Equation (v) we get,

$\frac{40}{{\mathrm{y}}}-\frac{15}{{\mathrm{x}}}-\frac{18}{{\mathrm{y}}}+\frac{15}{{\mathrm{x}}}$=25-(-6)

$\frac{40}{{\mathrm{y}}} -\frac{18}{{\mathrm{y}}}$ = 25+6

$\Rightarrow \frac{40 – 18}{y} = 31$

$\Rightarrow \frac{22}{y} = 31$

$\Rightarrow \frac{22}{31} = y$

$\Rightarrow y = \frac{22}{31}$

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

$\frac{8}{y} – \frac{3}{x} = 5$

$\Rightarrow \frac{8}{\left(\frac{22}{31}\right)} – \frac{3}{x} = 5$

$\Rightarrow \frac{8 \times 31}{22} – \frac{3}{x} = 5$

$\Rightarrow \frac{248}{22} – \frac{3}{x} = 5$

$\Rightarrow \frac{248}{22} – 5 = \frac{3}{x}$

$\Rightarrow \frac{248 – 110}{22} = \frac{3}{x}$

$\Rightarrow \frac{138}{22} = \frac{3}{x}$

$\Rightarrow 138x = 22 \times 3$

$\Rightarrow x = \frac{22 \times 3}{138}$

$\Rightarrow x = \frac{22}{46}$

$\Rightarrow x = \frac{11}{23}$

Required Solutions are x = $\frac{11}{23}$ and y = $\frac{22}{31}$