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Expression Formation

Solve the fol...

Question

Solve the following equation.
If $x < 0, y < 0, x + y + x / y = 1 / 2$ and $(x + y) (x / y) = - 1 / 2$ , then $x =$ and $y =$ ____.

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Solution

Put $x + y = u$ , $x / y = v, u + v = 1 / 2$ , $u v = - 1 / 2$
then u and v are roots of $t^{2} - \frac{1}{2} t - \frac{1}{2} = 0$
or $2 t^{2} - t - 1 = 0$
Hence $u = 1, v = - 1 / 2$ or $u = - 1 / 2$ , $v = 1$
$∴ x + y = 1$ , $x / y = - 1 / 2$
or $x + y = - 1 / 2$ , $x / y = 1$
Solving these, we shall get
$x = - 1, y = 2$ or $x = y = - 1 / 4$
Since $x < 0, y < 0$ , we get the answer $x = y = - 1 / 4$ .

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