Решение заданий 20 и 21

Задание 20. Представьте в виде дроби выражение:

1. $$ \frac{5b}{28} + \frac{9b}{28} = \frac{5b+9b}{28} = \frac{14b}{28} = \frac{b}{2} $$
2. $$ \frac{9m}{7n} - \frac{2m}{7n} = \frac{9m-2m}{7n} = \frac{7m}{7n} = \frac{m}{n} $$
3. $$ \frac{5x-3y}{8z} + \frac{3x-13y}{8z} = \frac{5x-3y+3x-13y}{8z} = \frac{8x-16y}{8z} = \frac{8(x-2y)}{8z} = \frac{x-2y}{z} $$

Задание 21. Упростите выражение:

1. $$ \frac{a-2}{a-1} - \frac{a}{1-a} = \frac{a-2}{a-1} + \frac{a}{a-1} = \frac{a-2+a}{a-1} = \frac{2a-2}{a-1} = \frac{2(a-1)}{a-1} = 2 $$
2. $$ \frac{3y+7}{4-y} + \frac{y+15}{y-4} = \frac{3y+7}{4-y} - \frac{y+15}{4-y} = \frac{3y+7-(y+15)}{4-y} = \frac{3y+7-y-15}{4-y} = \frac{2y-8}{4-y} = \frac{2(y-4)}{4-y} = -2 $$
3. $$ \frac{(2a-3)^2}{9a-27} + \frac{(a-6)^2}{27-9a} = \frac{(2a-3)^2}{9(a-3)} - \frac{(a-6)^2}{9(a-3)} = \frac{4a^2 - 12a + 9 - (a^2 - 12a + 36)}{9(a-3)} = \frac{3a^2 - 27}{9(a-3)} = \frac{3(a^2 - 9)}{9(a-3)} = \frac{3(a-3)(a+3)}{9(a-3)} = \frac{a+3}{3} $$
4. $$ \frac{25-3x}{(x-5)^2} - \frac{7x- x^2}{(5-x)^2} = \frac{25-3x}{(x-5)^2} - \frac{7x- x^2}{(x-5)^2} = \frac{25-3x - (7x-x^2)}{(x-5)^2} = \frac{25-3x - 7x+x^2}{(x-5)^2} = \frac{x^2-10x+25}{(x-5)^2} = \frac{(x-5)^2}{(x-5)^2} = 1 $$
5. $$ \frac{4c-3d}{cd} - \frac{c-3d}{cd} = \frac{4c-3d-(c-3d)}{cd} = \frac{4c-3d-c+3d}{cd} = \frac{3c}{cd} = \frac{3}{d} $$
6. $$ \frac{6x}{x^2-16} - \frac{24}{x^2-16} = \frac{6x-24}{x^2-16} = \frac{6(x-4)}{(x-4)(x+4)} = \frac{6}{x+4} $$
7. $$ \frac{m^2 +10m}{9-m^2} - \frac{4m-9}{9-m^2} = \frac{m^2 +10m - (4m-9)}{9-m^2} = \frac{m^2 +10m - 4m + 9}{9-m^2} = \frac{m^2 +6m + 9}{9-m^2} = \frac{(m+3)^2}{(3-m)(3+m)} = \frac{m+3}{3-m} $$