Задание 6. Выполните действие. А) 1) 1/7 + 3/7; 2) 11/15 − 7/15; 3) 3/7 + a/7 = (3+a)/7; 4) 2/9 + x/9; 5) 1/10 + 2x/10; 6) x/5 − 2/5; 7) a/3 + b/3; 8) 2a/11 − b/11; 9) y²/8 − 1/8; 10) 6/x + 2/x; 11) 3a/x − 4/x; 12) 2x²/y³ + x/y³; 13) 5b/a² − 3b/a²; 14) x/(2y) − a/(2y); 15) −2a/b² + a/b². Б) 1) (x+2)/3 + x/3; 2) (x+2)/3 − x/3; 3) (2a−3)/4 + a/4; 4) (2a−3)/4 − a/4; 5) (5b−a)/2 + a/2; 6) (5b−a)/2 − a/2; 7) (3−2x)/a + x/a; 8) (x²−y²)/a + 2x²/a; 9) (x²−y²)/a − 2x²/a; 10) (2a+3b−5)/a² + 5b/a²; 11) (2a+3b−5)/a² − 5b/a²; 12) (8−x²+2x)/(3a) + 3x²/(3a); 13) (8−x²+2x)/(3a) − 3x²/(3a); 14) (15+x+y+b)/(4ab) + 7b/(4ab). В) 1) 5/7 − (2+x)/7 = (5−(2+x))/7 = (3−x)/7; 2) 2/3 − (1−x)/3; 3) x/2 − (2x+3)/2; 4) a/5 − (9−a)/5; 5) b/y² − (b−a)/y²; 6) a/(2x) − (a+b)/(2x); 7) m/(6x) − (m+a)/(6x); 8) 2a/(11b) − (a+2)/(11b); 9) 2/(3a) − (3b+4)/(3a); 10) (x+2)/(3y) − (2x+5)/(3y); 11) (a−7)/a² − (15+2a)/a²; 12) (x+1)/(2y) − (3x−6)/(2y); 13) (5x−2y)/a − (3x+y)/a; 14) (12a+5b)/(3b) − (2a−b)/(3b); 15) (14m−n)/(xy) − (2n−3m)/(xy); 16) (3a³+x)/(4a) − (5x−2a³)/(4a). Г) 1) 2/(x−3) + a/(3−x) = 2/(x−3) − a/(x−3) = (2−a)/(x−3); 2) 1/(a−2) + b/(2−a); 3) m/(b−y) − 8/(y−b); 4) 5x/(a−b) − 2x/(b−a); 5) 3y/(x−a) + 2y/(a−x); 6) (a+5)/(b−4) + a/(4−b); 7) (x²+2)/(y−5) + 2/(5−y); 8) 3a/(2b−5) − 4a/(5−2b); 9) 2/(a−1) − 4/(−a+1); 10) b/(x−3) + a/(−x+3).‌

При одинаковых знаменателях складываем или вычитаем числители, сохраняя знаменатель. В пунктах, где знаменатели противоположны, используем равенство вида \(c-d=-(d-c)\).

А

1. \(\frac{1}{7}+\frac{3}{7}=\frac{1+3}{7}=\frac{4}{7}\).
2. \(\frac{11}{15}-\frac{7}{15}=\frac{11-7}{15}=\frac{4}{15}\).
3. \(\frac{3}{7}+\frac{a}{7}=\frac{3+a}{7}\).
4. \(\frac{2}{9}+\frac{x}{9}=\frac{x+2}{9}\).
5. \(\frac{1}{10}+\frac{2x}{10}=\frac{1+2x}{10}\).
6. \(\frac{x}{5}-\frac{2}{5}=\frac{x-2}{5}\).
7. \(\frac{a}{3}+\frac{b}{3}=\frac{a+b}{3}\).
8. \(\frac{2a}{11}-\frac{b}{11}=\frac{2a-b}{11}\).
9. \(\frac{y^2}{8}-\frac{1}{8}=\frac{y^2-1}{8}\).
10. \(\frac{6}{x}+\frac{2}{x}=\frac{8}{x}\).
11. \(\frac{3a}{x}-\frac{4}{x}=\frac{3a-4}{x}\).
12. \(\frac{2x^2}{y^3}+\frac{x}{y^3}=\frac{2x^2+x}{y^3}\).
13. \(\frac{5b}{a^2}-\frac{3b}{a^2}=\frac{2b}{a^2}\).
14. \(\frac{x}{2y}-\frac{a}{2y}=\frac{x-a}{2y}\).
15. \(-\frac{2a}{b^2}+\frac{a}{b^2}=-\frac{a}{b^2}\).

Б

1. \(\frac{x+2}{3}+\frac{x}{3}=\frac{2x+2}{3}\).
2. \(\frac{x+2}{3}-\frac{x}{3}=\frac{2}{3}\).
3. \(\frac{2a-3}{4}+\frac{a}{4}=\frac{3a-3}{4}\).
4. \(\frac{2a-3}{4}-\frac{a}{4}=\frac{a-3}{4}\).
5. \(\frac{5b-a}{2}+\frac{a}{2}=\frac{5b}{2}\).
6. \(\frac{5b-a}{2}-\frac{a}{2}=\frac{5b-2a}{2}\).
7. \(\frac{3-2x}{a}+\frac{x}{a}=\frac{3-x}{a}\).
8. \(\frac{x^2-y^2}{a}+\frac{2x^2}{a}=\frac{3x^2-y^2}{a}\).
9. \(\frac{x^2-y^2}{a}-\frac{2x^2}{a}=\frac{-x^2-y^2}{a}\).
10. \(\frac{2a+3b-5}{a^2}+\frac{5b}{a^2}=\frac{2a+8b-5}{a^2}\).
11. \(\frac{2a+3b-5}{a^2}-\frac{5b}{a^2}=\frac{2a-5}{a^2}\).
12. \(\frac{8-x^2+2x}{3a}+\frac{3x^2}{3a}=\frac{8+2x+2x^2}{3a}\).
13. \(\frac{8-x^2+2x}{3a}-\frac{3x^2}{3a}=\frac{8+2x-4x^2}{3a}\).
14. \(\frac{15+x+y+b}{4ab}+\frac{7b}{4ab}=\frac{15+x+y+8b}{4ab}\).

В

1. \(\frac{5}{7}-\frac{2+x}{7}=\frac{5-(2+x)}{7}=\frac{3-x}{7}\).
2. \(\frac{2}{3}-\frac{1-x}{3}=\frac{2-(1-x)}{3}=\frac{1+x}{3}\).
3. \(\frac{x}{2}-\frac{2x+3}{2}=\frac{-x-3}{2}\).
4. \(\frac{a}{5}-\frac{9-a}{5}=\frac{2a-9}{5}\).
5. \(\frac{b}{y^2}-\frac{b-a}{y^2}=\frac{a}{y^2}\).
6. \(\frac{a}{2x}-\frac{a+b}{2x}=-\frac{b}{2x}\).
7. \(\frac{m}{6x}-\frac{m+a}{6x}=-\frac{a}{6x}\).
8. \(\frac{2a}{11b}-\frac{a+2}{11b}=\frac{a-2}{11b}\).
9. \(\frac{2}{3a}-\frac{3b+4}{3a}=\frac{-3b-2}{3a}\).
10. \(\frac{x+2}{3y}-\frac{2x+5}{3y}=\frac{-x-3}{3y}\).
11. \(\frac{a-7}{a^2}-\frac{15+2a}{a^2}=\frac{-a-22}{a^2}\).
12. \(\frac{x+1}{2y}-\frac{3x-6}{2y}=\frac{-2x+7}{2y}\).
13. \(\frac{5x-2y}{a}-\frac{3x+y}{a}=\frac{2x-3y}{a}\).
14. \(\frac{12a+5b}{3b}-\frac{2a-b}{3b}=\frac{10a+6b}{3b}\).
15. \(\frac{14m-n}{xy}-\frac{2n-3m}{xy}=\frac{17m-3n}{xy}\).
16. \(\frac{3a^3+x}{4a}-\frac{5x-2a^3}{4a}=\frac{5a^3-4x}{4a}\).

Г

1. Так как \(3-x=-(x-3)\), получаем \(\frac{2}{x-3}+\frac{a}{3-x}=\frac{2}{x-3}-\frac{a}{x-3}=\frac{2-a}{x-3}\).
2. Так как \(2-a=-(a-2)\), \(\frac{1}{a-2}+\frac{b}{2-a}=\frac{1-b}{a-2}\).
3. Так как \(y-b=-(b-y)\), \(\frac{m}{b-y}-\frac{8}{y-b}=\frac{m+8}{b-y}\).
4. Так как \(b-a=-(a-b)\), \(\frac{5x}{a-b}-\frac{2x}{b-a}=\frac{7x}{a-b}\).
5. Так как \(a-x=-(x-a)\), \(\frac{3y}{x-a}+\frac{2y}{a-x}=\frac{y}{x-a}\).
6. Так как \(4-b=-(b-4)\), \(\frac{a+5}{b-4}+\frac{a}{4-b}=\frac{5}{b-4}\).
7. Так как \(5-y=-(y-5)\), \(\frac{x^2+2}{y-5}+\frac{2}{5-y}=\frac{x^2}{y-5}\).
8. Так как \(5-2b=-(2b-5)\), \(\frac{3a}{2b-5}-\frac{4a}{5-2b}=\frac{7a}{2b-5}\).
9. Поскольку \(-a+1=-(a-1)\), \(\frac{2}{a-1}-\frac{4}{-a+1}=\frac{6}{a-1}\).
10. Поскольку \(-x+3=-(x-3)\), \(\frac{b}{x-3}+\frac{a}{-x+3}=\frac{b-a}{x-3}\).