Упр.1070 ГДЗ Мерзляк Полонский 8 класс (Алгебра)

1) (2a-1)/(a-4) — (3a+2)/2(a-4);
2) (x+2)/(3x+9) — (4-x)/(5x+15);
3) (m+1)/(m-3) — (m+2)/(m+3);

1. $$\frac{2a-1}{a-4}-\frac{3a+2}{2(a-4)}=\frac{2(2a-1)-(3a+2)}{2(a-4)}=\frac{4a-2-3a-2}{2(a-4)}=\frac{a-4}{2(a-4)}=\frac12.$$

2. $$\frac{x+2}{3x+9}-\frac{4-x}{5x+15}=\frac{x+2}{3(x+3)}-\frac{4-x}{5(x+3)}=\frac{5(x+2)-3(4-x)}{15(x+3)}$$
   $$=\frac{5x+10-12+3x}{15(x+3)}=\frac{8x-2}{15(x+3)}.$$

3. $$\frac{m+1}{m-3}-\frac{m+2}{m+3}=\frac{(m+1)(m+3)-(m+2)(m-3)}{(m-3)(m+3)}$$
   $$=\frac{m^2+4m+3-(m^2-m-6)}{m^2-9}=\frac{5m+9}{m^2-9}.$$

4. $$\frac{x}{x+y}-\frac{2y^2}{y^2-x^2}-\frac{y}{x-y}=\frac{x}{x+y}+\frac{2y^2}{(x-y)(x+y)}-\frac{y}{x-y}$$
   $$=\frac{x(x-y)+2y^2-y(x+y)}{(x+y)(x-y)}=\frac{x^2-xy+2y^2-xy-y^2}{x^2-y^2}$$
   $$=\frac{x^2-2xy+y^2}{x^2-y^2}=\frac{(x-y)^2}{(x-y)(x+y)}=\frac{x-y}{x+y}.$$

5. $$\frac{m}{3m-2n}-\frac{3m^2-3mn}{9m^2-12mn+4n^2}=\frac{m}{3m-2n}-\frac{3m^2-3mn}{(3m-2n)^2}$$
   $$=\frac{m(3m-2n)-(3m^2-3mn)}{(3m-2n)^2}=\frac{3m^2-2mn-3m^2+3mn}{(3m-2n)^2}=\frac{mn}{(3m-2n)^2}.$$

6. $$\frac{a+3}{a^2-2a}-\frac{a-2}{5a-10}+\frac{a+2}{5a}=\frac{a+3}{a(a-2)}-\frac{a-2}{5(a-2)}+\frac{a+2}{5a}$$
   $$=\frac{5(a+3)-a(a-2)+(a+2)(a-2)}{5a(a-2)}$$
   $$=\frac{5a+15-a^2+2a+a^2-4}{5a(a-2)}=\frac{7a+11}{5a(a-2)}.$$

7. $$2-\frac{2}{m-2}\cdot m=\frac{2(m-2)-2m}{m-2}=\frac{2m-4-2m}{m-2}=\frac{-4}{m-2}.$$

Ответ

1) $$\frac12$$; 2) $$\frac{8x-2}{15(x+3)}$$; 3) $$\frac{5m+9}{m^2-9}$$; 4) $$\frac{x-y}{x+y}$$; 5) $$\frac{mn}{(3m-2n)^2}$$; 6) $$\frac{7a+11}{5a(a-2)}$$; 7) $$\frac{-4}{m-2}$$.