1. Выполните сложение или вычитание дробей: 1) a) x/7 + y/7; б) m/2 - n/2; в) a/b + 2a/b; г) 3x/y - x/y; д) (a+5b)/15 + (2a+4b)/15; e) (b+c)/3a - (b-2c)/3a; ж) (3x+2y)/xy + (2y-5x)/xy; 2) a) (5x-7)/6x - (x-3)/6x + (2x-8)/6x; б) (8y-5)/7y - (2y-1)/7y - (10-y)/7y; в) (x-5)/(x²-49) + 12/(x²-49); г) (y²+2y)/(y²-4y+4) - 4y/(y²-4y+4); д) 3z/(z²-2z) - (8-z)/(z²-2z); 3) a) (a+3)/(a-1) - a/(1-a); б) (3x+2y)/(2x-3y) - (x-8y)/(3y-2x); в) b²/(2b-10) + 25/(10-2b); 4) a) (9y+1)/(y²-4) - (y-8)/(4-y²) + (1-7y)/(y²-4); б) 3x/(x³-1) - (4x-1)/(1-x³) - x²/(1-x³).

1. Сложение и вычитание дробей с одинаковыми знаменателями:

1. а)

   \[ \frac{x}{7} + \frac{y}{7} = \frac{x+y}{7} \]

2. б)

   \[ \frac{m}{2} - \frac{n}{2} = \frac{m-n}{2} \]

3. в)

   \[ \frac{a}{b} + \frac{2a}{b} = \frac{a+2a}{b} = \frac{3a}{b} \]

4. г)

   \[ \frac{3x}{y} - \frac{x}{y} = \frac{3x-x}{y} = \frac{2x}{y} \]

5. д)

   \[ \frac{a+5b}{15} + \frac{2a+4b}{15} = \frac{a+5b+2a+4b}{15} = \frac{3a+9b}{15} = \frac{a+3b}{5} \]

6. е)

   \[ \frac{b+c}{3a} - \frac{b-2c}{3a} = \frac{b+c - (b-2c)}{3a} = \frac{b+c-b+2c}{3a} = \frac{3c}{3a} = \frac{c}{a} \]

7. ж)

   \[ \frac{3x+2y}{xy} + \frac{2y-5x}{xy} = \frac{3x+2y+2y-5x}{xy} = \frac{4y-2x}{xy} \]

8. 2) а)

   \[ \frac{5x-7}{6x} - \frac{x-3}{6x} + \frac{2x-8}{6x} = \frac{5x-7 - (x-3) + 2x-8}{6x} = \frac{5x-7-x+3+2x-8}{6x} = \frac{6x-12}{6x} = \frac{x-2}{x} \]

9. б)

   \[ \frac{8y-5}{7y} - \frac{2y-1}{7y} - \frac{10-y}{7y} = \frac{8y-5 - (2y-1) - (10-y)}{7y} = \frac{8y-5-2y+1-10+y}{7y} = \frac{7y-14}{7y} = \frac{y-2}{y} \]

10. в)

    \[ \frac{x-5}{x^2-49} + \frac{12}{x^2-49} = \frac{x-5+12}{x^2-49} = \frac{x+7}{x^2-49} = \frac{x+7}{(x-7)(x+7)} = \frac{1}{x-7} \]

11. г)

    \[ \frac{y^2+2y}{y^2-4y+4} - \frac{4y}{y^2-4y+4} = \frac{y^2+2y-4y}{y^2-4y+4} = \frac{y^2-2y}{y^2-4y+4} = \frac{y(y-2)}{(y-2)^2} = \frac{y}{y-2} \]

12. д)

    \[ \frac{3z}{z^2-2z} - \frac{8-z}{z^2-2z} = \frac{3z - (8-z)}{z^2-2z} = \frac{3z-8+z}{z^2-2z} = \frac{4z-8}{z(z-2)} = \frac{4(z-2)}{z(z-2)} = \frac{4}{z} \]

13. 3) а)

    \[ \frac{a+3}{a-1} - \frac{a}{1-a} = \frac{a+3}{a-1} - \frac{-a}{a-1} = \frac{a+3+a}{a-1} = \frac{2a+3}{a-1} \]

14. б)

    \[ \frac{3x+2y}{2x-3y} - \frac{x-8y}{3y-2x} = \frac{3x+2y}{2x-3y} - \frac{x-8y}{-(2x-3y)} = \frac{3x+2y}{2x-3y} + \frac{x-8y}{2x-3y} = \frac{3x+2y+x-8y}{2x-3y} = \frac{4x-6y}{2x-3y} = \frac{2(2x-3y)}{2x-3y} = 2 \]

15. в)

    \[ \frac{b^2}{2b-10} + \frac{25}{10-2b} = \frac{b^2}{2b-10} + \frac{25}{-(2b-10)} = \frac{b^2}{2b-10} - \frac{25}{2b-10} = \frac{b^2-25}{2b-10} = \frac{(b-5)(b+5)}{2(b-5)} = \frac{b+5}{2} \]

16. 4) а)

    \[ \frac{9y+1}{y^2-4} - \frac{y-8}{4-y^2} + \frac{1-7y}{y^2-4} = \frac{9y+1}{y^2-4} - \frac{y-8}{-(y^2-4)} + \frac{1-7y}{y^2-4} = \frac{9y+1}{y^2-4} + \frac{y-8}{y^2-4} + \frac{1-7y}{y^2-4} = \frac{9y+1+y-8+1-7y}{y^2-4} = \frac{3y-6}{y^2-4} = \frac{3(y-2)}{(y-2)(y+2)} = \frac{3}{y+2} \]

17. б)

    \[ \frac{3x}{x^3-1} - \frac{4x-1}{1-x^3} - \frac{x^2}{1-x^3} = \frac{3x}{x^3-1} - \frac{4x-1}{-(x^3-1)} - \frac{x^2}{-(x^3-1)} = \frac{3x}{x^3-1} + \frac{4x-1}{x^3-1} + \frac{x^2}{x^3-1} = \frac{3x+4x-1+x^2}{x^3-1} = \frac{x^2+7x-1}{x^3-1} \]