Упр.926 ГДЗ Никольский Потапов 9 класс (Алгебра)

1. а) $$\left(\frac{1}{x}+\frac{1}{y}\right)(x-y)+(x+y)\left(\frac{1}{x}-\frac{1}{y}\right)$$; б) $$\left(\frac{y}{x}-\frac{x}{y}\right):\left(2-\frac{x}{y}-\frac{y}{x}\right):\left(\frac{y}{x}+1\right)$$.

а)

$$\left(\frac{1}{x}+\frac{1}{y}\right)(x-y)+\left(x+y\right)\left(\frac{1}{x}-\frac{1}{y}\right)$$
$$=\left(\frac{x-y}{x}+\frac{x-y}{y}\right)+\left(\frac{x+y}{x}-\frac{x+y}{y}\right)$$
$$= \left(1-\frac{y}{x}+\frac{x}{y}-1\right)+\left(1-\frac{x}{y}+\frac{y}{x}-1\right)=0.$$

б)

$$\left(\frac{y}{x}-\frac{x}{y}\right):\left(2-\frac{x}{y}-\frac{y}{x}\right):\left(\frac{y}{x}+1\right)$$
$$=\frac{y^2-x^2}{xy}:\frac{2xy-x^2-y^2}{xy}:\frac{y+x}{x}$$
$$=\frac{(y-x)(y+x)}{xy}:\frac{-(x-y)^2}{xy}:\frac{y+x}{x}$$
$$=\frac{(y-x)(y+x)}{xy}\cdot\frac{xy}{-(x-y)^2}\cdot\frac{x}{y+x} =\frac{x}{y-x} =\frac{x}{x-y}.$$

Ответ

а) $$0$$; б) $$\frac{x}{x-y}$$.