8 класс. Сложение и вычитание дробей с разными знаменателями

### Блок 1 1) $\frac{7-3y}{y} - \frac{8-3x}{x} = \frac{x(7-3y) - y(8-3x)}{xy} = \frac{7x-3xy-8y+3xy}{xy} = \frac{7x-8y}{xy}$ 2) $\frac{3x+2}{5x} - \frac{5x+3y}{10xy} - \frac{y-1}{2y} = \frac{2y(3x+2) - (5x+3y) - 5x(y-1)}{10xy} = \frac{6xy+4y-5x-3y-5xy+5x}{10xy} = \frac{xy+y}{10xy} = \frac{y(x+1)}{10xy} = \frac{x+1}{10x}$ 3) $\frac{c-2}{3(c+4)} + \frac{c}{c+4} = \frac{c-2+3c}{3(c+4)} = \frac{4c-2}{3(c+4)}$ 4) $\frac{4}{c^2-9} - \frac{2}{c^2+3c} = \frac{4}{(c-3)(c+3)} - \frac{2}{c(c+3)} = \frac{4c-2(c-3)}{c(c-3)(c+3)} = \frac{4c-2c+6}{c(c^2-9)} = \frac{2c+6}{c(c^2-9)} = \frac{2(c+3)}{c(c-3)(c+3)} = \frac{2}{c^2-3c}$ 5) $\frac{7}{x} - 2x = \frac{7-2x^2}{x}$ ### Блок 2 1) $\frac{m-n}{m^2} - \frac{n-m}{m-n} = \frac{m-n}{m^2} - \frac{-(m-n)}{m-n} = \frac{m-n}{m^2} - (-1) = \frac{m-n+m^2}{m^2}$ 2) $\frac{(a-b)^2}{18b} - \frac{(a-b)^2}{12b} + \frac{a^2-b^2}{36b} = \frac{2(a-b)^2 - 3(a-b)^2 + (a^2-b^2)}{36b} = \frac{-(a-b)^2 + a^2-b^2}{36b} = \frac{-(a^2-2ab+b^2) + a^2-b^2}{36b} = \frac{-a^2+2ab-b^2+a^2-b^2}{36b} = \frac{2ab-2b^2}{36b} = \frac{2b(a-b)}{36b} = \frac{a-b}{18}$ 3) $\frac{b-2}{2b-6} - \frac{b-1}{3b-9} = \frac{b-2}{2(b-3)} - \frac{b-1}{3(b-3)} = \frac{3(b-2)-2(b-1)}{6(b-3)} = \frac{3b-6-2b+2}{6(b-3)} = \frac{b-4}{6(b-3)}$ 4) $\frac{3a(x-9a)}{x(x-3a)} - \frac{3a^2-x^2}{a(x-3a)} = \frac{3a^2(x-9a) - x(3a^2-x^2)}{ax(x-3a)} = \frac{3a^2x-27a^3-3a^2x+x^3}{ax(x-3a)} = \frac{x^3-27a^3}{ax(x-3a)} = \frac{(x-3a)(x^2+3ax+9a^2)}{ax(x-3a)} = \frac{x^2+3ax+9a^2}{ax}$ 5) $\frac{15b}{5-b} - 3b = \frac{15b - 3b(5-b)}{5-b} = \frac{15b-15b+3b^2}{5-b} = \frac{3b^2}{5-b}$