Question

If $l=\frac{y-1}{x}, \ m=\frac{1-x}{y}$ and n = x – y, show that l + m + n + lmn = 0.

Answer 1

Given that :

$l = \frac{y - 1}{x} \: m = \frac{1 - x}{y} \: and \: n = x - y$

Now, we have to show that :

$l + m + n + lmn = 0$

On taking LHS :

$l + m + n + lmn \\ \\ = > \frac{y - 1}{x} + \frac{1 - x}{y} + x - y + ( (\frac{y - 1}{x} )\times (\frac{1 - x}{y} )\times( x - y)) \\ \\ = > (\frac{ {y}^{2} - y + x - {x}^{2} + {x}^{2} y - x {y}^{2} }{xy}) + (\frac{(y - xy - 1 + x)(x - y)}{xy} ) \\ \\ = > \frac{ {y}^{2} - y + x - {x}^{2} + {x}^{2}y - x {y}^{2} }{xy} + \frac{xy - {y}^{2} - {x}^{2}y + x {y}^{2} - x + y + {x}^{2} - xy }{xy} \\ \\ = > \frac{ {y}^{2} - y + x - {x}^{2} + {x}^{2}y - x {y}^{2} - {y}^{2} - {x}^{2}y + x {y}^{2} - x + y + {x}^{2} }{xy} \\ \\ = > \frac{0}{xy} \\ \\ = > 0 = RHS$

HENCE PROVED