Страница 39 номер 2.3.6, ГДЗ по алгебре за 7, 8 и 9 класс к задачнику Ткачевой
Выполнить действия: 1) \(\frac{2a - b}{2c} + \frac{6a + b}{2c}\); 2) \(\frac{a + 3b}{3c} + \frac{6b - a}{3c}\); 3) \(\frac{1 - m}{m} - \frac{1 - n}{n}\); 4) \(\frac{2m - 3}{m} - \frac{3n - 2}{n}\); 5) \(\frac{5 + x}{2x^2} + \frac{1 - 2x}{x}\); 6) \(\frac{2 - x}{y} + \frac{xy - 3}{3y^2}\); 7) \(\frac{2b - 5}{6a} - \frac{b + 1}{8}\); 8) \(\frac{a - 1}{14} - \frac{2 - 5a}{21b}\); 9) \(\frac{6a}{11b} - 4 - a\); 10) \(\frac{3b}{13a} - 2 + b\); 11) \(\frac{3}{4x^2 y^3} + \frac{1}{6x^3 y^2}\); 12) \(\frac{2}{15x^3 y^4} + \frac{3}{25x^4 y^3}\); 13) \(\frac{a + 2}{5(a - 1)} - \frac{2a - 1}{3(a - 1)}\); 14) \(\frac{2b + 1}{6(b - 2)} - \frac{b - 1}{5(b - 2)}\); 15) \(\frac{3}{2(m - n)} + \frac{4}{3(n - m)}\); 16) \(\frac{2}{3(p - q)} - \frac{3}{4(q - p)}\); 17) \(\frac{3}{ab + b^2} - \frac{a - 1}{a^2 b + ab^2}\); 18) \(\frac{4}{x^2 + xy} - \frac{x - 2}{xy^2 + x^2 y}\); 19) \(\frac{2x - 1}{x^2 - 9} + \frac{3}{3 - x}\); 20) \(\frac{5}{2 - y} - \frac{y - 3}{y^2 - 4}\); 21) \(\frac{3a - b}{a^2 + 2ab + b^2} + \frac{2}{a + b}\); 22) \(\frac{3}{a - b} + \frac{2a - 3b}{a^2 - 2ab + b^2}\); 23) \(\frac{2}{m^2 - n^2} - \frac{1}{m^2 + 2mn + n^2}\); 24) \(\frac{2}{x^2 - 2xy + y^2} - \frac{3}{x^2 - y^2}\).
1) \(\frac{2a - b}{2c} + \frac{6a + b}{2c} = \frac{2a - b + 6a + b}{2c} = \frac{8a}{2c} = \frac{4a}{c}\);
2) \(\frac{a + 3b}{3c} + \frac{6b - a}{3c} = \frac{a + 3b + 6b - a}{3c} = \frac{9b}{3c} = \frac{3b}{c}\);
3) \(\frac{1 - m}{m} - \frac{1 - n}{n} = \frac{n(1 - m) - m(1 - n)}{mn} = \frac{n - mn - m + mn}{mn} = \frac{n - m}{mn}\);
4) \(\frac{2m - 3}{m} - \frac{3n - 2}{n} = \frac{n(2m - 3) - m(3n - 2)}{mn} = \frac{2mn - 3n - 3mn + 2m}{mn} = \frac{2m - mn - 3n}{mn}\);
5) \(\frac{5 + x}{2x^2} + \frac{1 - 2x}{x} = \frac{5 + x + 2x(1 - 2x)}{2x^2} = \frac{5 + x + 2x - 4x^2}{2x^2} = \frac{5 + 3x - 4x^2}{2x^2}\);
6) \(\frac{2 - x}{y} + \frac{xy - 3}{3y^2} = \frac{3y(2 - x) + xy - 3}{3y^2} = \frac{6y - 3xy + xy - 3}{3y^2} = \frac{6y - 2xy - 3}{3y^2}\);
7) \(\frac{2b - 5}{6a} - \frac{b + 1}{8} = \frac{4(2b - 5) - 3a(b + 1)}{24a} = \frac{8b - 20 - 3ab - 3a}{24a}\);
8) \(\frac{a - 1}{14} - \frac{2 - 5a}{21b} = \frac{3b(a - 1) - 2(2 - 5a)}{42b} = \frac{3ab - 3b - 4 + 10a}{42b}\);
9) \(\frac{6a}{11b} - 4 - a = \frac{6a - 4 \cdot 11b - a \cdot 11b}{11b} = \frac{6a - 44b - 11ab}{11b}\);
10) \(\frac{3b}{13a} - 2 + b = \frac{3b - 2 \cdot 13a + b \cdot 13a}{13a} = \frac{3b - 26a + 13ab}{13a}\);
11) \(\frac{3}{4x^2y^3} + \frac{1}{6x^3y^2} = \frac{3 \cdot 3x + 1 \cdot 2y}{12x^3y^3} = \frac{9x + 2y}{12x^3y^3}\);
12) \(\frac{2}{15x^3y^4} + \frac{3}{25x^4y^3} = \frac{2 \cdot 5x + 3 \cdot 3y}{75x^4y^4} = \frac{10x + 9y}{75x^4y^4}\);
13) \(\frac{a + 2}{5(a - 1)} - \frac{2a - 1}{3(a - 1)} = \frac{3(a + 2) - 5(2a - 1)}{15(a - 1)} = \frac{3a + 6 - 10a + 5}{15(a - 1)} = \frac{11 - 7a}{15(a - 1)}\);
14) \(\frac{2b + 1}{6(b - 2)} - \frac{b - 1}{5(b - 2)} = \frac{5(2b + 1) - 6(b - 1)}{30(b - 2)} = \frac{10b + 5 - 6b + 6}{30(b - 2)} = \frac{4b + 11}{30(b - 2)}\);
15) \(\frac{3}{2(m - n)} + \frac{4}{3(n - m)} = \frac{3}{2(m - n)} - \frac{4}{3(m - n)} = \frac{9 - 8}{6(m - n)} = \frac{1}{6(m - n)}\);
16) \(\frac{2}{3(p - q)} - \frac{3}{4(q - p)} = \frac{2}{3(p - q)} + \frac{3}{4(p - q)} = \frac{8 + 9}{12(p - q)} = \frac{17}{12(p - q)}\);
17) \(\frac{3}{ab + b^2} - \frac{a - 1}{a^2b + ab^2} = \frac{3}{b(a + b)} - \frac{a - 1}{ab(a + b)} = \frac{3a - (a - 1)}{ab(a + b)} = \frac{2a + 1}{ab(a + b)}\);
18) \(\frac{4}{x^2 + xy} - \frac{x - 2}{xy^2 + x^2y} = \frac{4}{x(x + y)} - \frac{x - 2}{xy(x + y)} = \frac{4y - (x - 2)}{xy(x + y)} = \frac{4y - x + 2}{xy(x + y)}\);
19) \(\frac{2x - 1}{x^2 - 9} + \frac{3}{3 - x} = \frac{2x - 1}{(x - 3)(x + 3)} - \frac{3}{x - 3} = \frac{2x - 1 - 3(x + 3)}{(x - 3)(x + 3)} = \frac{-x - 10}{x^2 - 9} = \frac{x + 10}{9 - x^2}\);
20) \(\frac{5}{2 - y} - \frac{y - 3}{y^2 - 4} = -\frac{5}{y - 2} - \frac{y - 3}{(y - 2)(y + 2)} = \frac{-5(y + 2) - (y - 3)}{(y - 2)(y + 2)} = \frac{-6y - 7}{y^2 - 4} = \frac{6y + 7}{4 - y^2}\);
21) \(\frac{3a - b}{a^2 + 2ab + b^2} + \frac{2}{a + b} = \frac{3a - b}{(a + b)^2} + \frac{2}{a + b} = \frac{3a - b + 2(a + b)}{(a + b)^2} = \frac{5a + b}{(a + b)^2}\);
22) \(\frac{3}{a - b} + \frac{2a - 3b}{a^2 - 2ab + b^2} = \frac{3}{a - b} + \frac{2a - 3b}{(a - b)^2} = \frac{3(a - b) + 2a - 3b}{(a - b)^2} = \frac{5a - 6b}{(a - b)^2}\);
23) \(\frac{2}{m^2 - n^2} - \frac{1}{m^2 + 2mn + n^2} = \frac{2}{(m - n)(m + n)} - \frac{1}{(m + n)^2} = \frac{2(m + n) - (m - n)}{(m + n)^2(m - n)} = \frac{m + 3n}{(m + n)^2(m - n)}\);
24) \(\frac{2}{x^2 - 2xy + y^2} - \frac{3}{x^2 - y^2} = \frac{2}{(x - y)^2} - \frac{3}{(x - y)(x + y)} = \frac{2(x + y) - 3(x - y)}{(x - y)^2(x + y)} = \frac{5y - x}{(x - y)^2(x + y)}\).
Ответ: 1) \(\frac{4a}{c}\); 2) \(\frac{3b}{c}\); 3) \(\frac{n - m}{mn}\); 4) \(\frac{2m - mn - 3n}{mn}\); 5) \(\frac{5 + 3x - 4x^2}{2x^2}\); 6) \(\frac{6y - 2xy - 3}{3y^2}\); 7) \(\frac{8b - 20 - 3ab - 3a}{24a}\); 8) \(\frac{3ab - 3b - 4 + 10a}{42b}\); 9) \(\frac{6a - 44b - 11ab}{11b}\); 10) \(\frac{3b - 26a + 13ab}{13a}\); 11) \(\frac{9x + 2y}{12x^3y^3}\); 12) \(\frac{10x + 9y}{75x^4y^4}\); 13) \(\frac{11 - 7a}{15(a - 1)}\); 14) \(\frac{4b + 11}{30(b - 2)}\); 15) \(\frac{1}{6(m - n)}\); 16) \(\frac{17}{12(p - q)}\); 17) \(\frac{2a + 1}{ab(a + b)}\); 18) \(\frac{4y - x + 2}{xy(x + y)}\); 19) \(\frac{x + 10}{9 - x^2}\); 20) \(\frac{6y + 7}{4 - y^2}\); 21) \(\frac{5a + b}{(a + b)^2}\); 22) \(\frac{5a - 6b}{(a - b)^2}\); 23) \(\frac{m + 3n}{(m + n)^2(m - n)}\); 24) \(\frac{5y - x}{(x - y)^2(x + y)}\).
Выполнить действия: 1) \(\frac{2a - b}{2c} + \frac{6a + b}{2c}\); 2) \(\frac{a + 3b}{3c} + \frac{6b - a}{3c}\); 3) \(\frac{1 - m}{m} - \frac{1 - n}{n}\); 4) \(\frac{2m - 3}{m} - \frac{3n - 2}{n}\); 5) \(\frac{5 + x}{2x^2} + \frac{1 - 2x}{x}\); 6) \(\frac{2 - x}{y} + \frac{xy - 3}{3y^2}\); 7) \(\frac{2b - 5}{6a} - \frac{b + 1}{8}\); 8) \(\frac{a - 1}{14} - \frac{2 - 5a}{21b}\); 9) \(\frac{6a}{11b} - 4 - a\); 10) \(\frac{3b}{13a} - 2 + b\); 11) \(\frac{3}{4x^2 y^3} + \frac{1}{6x^3 y^2}\); 12) \(\frac{2}{15x^3 y^4} + \frac{3}{25x^4 y^3}\); 13) \(\frac{a + 2}{5(a - 1)} - \frac{2a - 1}{3(a - 1)}\); 14) \(\frac{2b + 1}{6(b - 2)} - \frac{b - 1}{5(b - 2)}\); 15) \(\frac{3}{2(m - n)} + \frac{4}{3(n - m)}\); 16) \(\frac{2}{3(p - q)} - \frac{3}{4(q - p)}\); 17) \(\frac{3}{ab + b^2} - \frac{a - 1}{a^2 b + ab^2}\); 18) \(\frac{4}{x^2 + xy} - \frac{x - 2}{xy^2 + x^2 y}\); 19) \(\frac{2x - 1}{x^2 - 9} + \frac{3}{3 - x}\); 20) \(\frac{5}{2 - y} - \frac{y - 3}{y^2 - 4}\); 21) \(\frac{3a - b}{a^2 + 2ab + b^2} + \frac{2}{a + b}\); 22) \(\frac{3}{a - b} + \frac{2a - 3b}{a^2 - 2ab + b^2}\); 23) \(\frac{2}{m^2 - n^2} - \frac{1}{m^2 + 2mn + n^2}\); 24) \(\frac{2}{x^2 - 2xy + y^2} - \frac{3}{x^2 - y^2}\).
Дроби с разными знаменателями приводим к общему знаменателю. Если знаменатели двух дробей противоположны, у одной из дробей меняем знак знаменателя и знак перед дробью.
1) \(\frac{2a - b}{2c} + \frac{6a + b}{2c} = \frac{2a - b + 6a + b}{2c} = \frac{8a}{2c} = \frac{4a}{c}\);
2) \(\frac{a + 3b}{3c} + \frac{6b - a}{3c} = \frac{a + 3b + 6b - a}{3c} = \frac{9b}{3c} = \frac{3b}{c}\);
3) \(\frac{1 - m}{m} - \frac{1 - n}{n} = \frac{n(1 - m) - m(1 - n)}{mn} = \frac{n - mn - m + mn}{mn} = \frac{n - m}{mn}\);
4) \(\frac{2m - 3}{m} - \frac{3n - 2}{n} = \frac{n(2m - 3) - m(3n - 2)}{mn} = \frac{2mn - 3n - 3mn + 2m}{mn} = \frac{2m - mn - 3n}{mn}\);
5) \(\frac{5 + x}{2x^2} + \frac{1 - 2x}{x} = \frac{5 + x + 2x(1 - 2x)}{2x^2} = \frac{5 + x + 2x - 4x^2}{2x^2} = \frac{5 + 3x - 4x^2}{2x^2}\);
6) \(\frac{2 - x}{y} + \frac{xy - 3}{3y^2} = \frac{3y(2 - x) + xy - 3}{3y^2} = \frac{6y - 3xy + xy - 3}{3y^2} = \frac{6y - 2xy - 3}{3y^2}\);
7) \(\frac{2b - 5}{6a} - \frac{b + 1}{8} = \frac{4(2b - 5) - 3a(b + 1)}{24a} = \frac{8b - 20 - 3ab - 3a}{24a}\);
8) \(\frac{a - 1}{14} - \frac{2 - 5a}{21b} = \frac{3b(a - 1) - 2(2 - 5a)}{42b} = \frac{3ab - 3b - 4 + 10a}{42b}\);
9) \(\frac{6a}{11b} - 4 - a = \frac{6a - 4 \cdot 11b - a \cdot 11b}{11b} = \frac{6a - 44b - 11ab}{11b}\);
10) \(\frac{3b}{13a} - 2 + b = \frac{3b - 2 \cdot 13a + b \cdot 13a}{13a} = \frac{3b - 26a + 13ab}{13a}\);
11) \(\frac{3}{4x^2y^3} + \frac{1}{6x^3y^2} = \frac{3 \cdot 3x + 1 \cdot 2y}{12x^3y^3} = \frac{9x + 2y}{12x^3y^3}\);
12) \(\frac{2}{15x^3y^4} + \frac{3}{25x^4y^3} = \frac{2 \cdot 5x + 3 \cdot 3y}{75x^4y^4} = \frac{10x + 9y}{75x^4y^4}\);
13) \(\frac{a + 2}{5(a - 1)} - \frac{2a - 1}{3(a - 1)} = \frac{3(a + 2) - 5(2a - 1)}{15(a - 1)} = \frac{3a + 6 - 10a + 5}{15(a - 1)} = \frac{11 - 7a}{15(a - 1)}\);
14) \(\frac{2b + 1}{6(b - 2)} - \frac{b - 1}{5(b - 2)} = \frac{5(2b + 1) - 6(b - 1)}{30(b - 2)} = \frac{10b + 5 - 6b + 6}{30(b - 2)} = \frac{4b + 11}{30(b - 2)}\);
15) \(\frac{3}{2(m - n)} + \frac{4}{3(n - m)} = \frac{3}{2(m - n)} - \frac{4}{3(m - n)} = \frac{9 - 8}{6(m - n)} = \frac{1}{6(m - n)}\);
16) \(\frac{2}{3(p - q)} - \frac{3}{4(q - p)} = \frac{2}{3(p - q)} + \frac{3}{4(p - q)} = \frac{8 + 9}{12(p - q)} = \frac{17}{12(p - q)}\);
17) \(\frac{3}{ab + b^2} - \frac{a - 1}{a^2b + ab^2} = \frac{3}{b(a + b)} - \frac{a - 1}{ab(a + b)} = \frac{3a - (a - 1)}{ab(a + b)} = \frac{2a + 1}{ab(a + b)}\);
18) \(\frac{4}{x^2 + xy} - \frac{x - 2}{xy^2 + x^2y} = \frac{4}{x(x + y)} - \frac{x - 2}{xy(x + y)} = \frac{4y - (x - 2)}{xy(x + y)} = \frac{4y - x + 2}{xy(x + y)}\);
19) \(\frac{2x - 1}{x^2 - 9} + \frac{3}{3 - x} = \frac{2x - 1}{(x - 3)(x + 3)} - \frac{3}{x - 3} = \frac{2x - 1 - 3(x + 3)}{(x - 3)(x + 3)} = \frac{-x - 10}{x^2 - 9} = \frac{x + 10}{9 - x^2}\);
20) \(\frac{5}{2 - y} - \frac{y - 3}{y^2 - 4} = -\frac{5}{y - 2} - \frac{y - 3}{(y - 2)(y + 2)} = \frac{-5(y + 2) - (y - 3)}{(y - 2)(y + 2)} = \frac{-6y - 7}{y^2 - 4} = \frac{6y + 7}{4 - y^2}\);
21) \(\frac{3a - b}{a^2 + 2ab + b^2} + \frac{2}{a + b} = \frac{3a - b}{(a + b)^2} + \frac{2}{a + b} = \frac{3a - b + 2(a + b)}{(a + b)^2} = \frac{5a + b}{(a + b)^2}\);
22) \(\frac{3}{a - b} + \frac{2a - 3b}{a^2 - 2ab + b^2} = \frac{3}{a - b} + \frac{2a - 3b}{(a - b)^2} = \frac{3(a - b) + 2a - 3b}{(a - b)^2} = \frac{5a - 6b}{(a - b)^2}\);
23) \(\frac{2}{m^2 - n^2} - \frac{1}{m^2 + 2mn + n^2} = \frac{2}{(m - n)(m + n)} - \frac{1}{(m + n)^2} = \frac{2(m + n) - (m - n)}{(m + n)^2(m - n)} = \frac{m + 3n}{(m + n)^2(m - n)}\);
24) \(\frac{2}{x^2 - 2xy + y^2} - \frac{3}{x^2 - y^2} = \frac{2}{(x - y)^2} - \frac{3}{(x - y)(x + y)} = \frac{2(x + y) - 3(x - y)}{(x - y)^2(x + y)} = \frac{5y - x}{(x - y)^2(x + y)}\).
Ответ: 1) \(\frac{4a}{c}\); 2) \(\frac{3b}{c}\); 3) \(\frac{n - m}{mn}\); 4) \(\frac{2m - mn - 3n}{mn}\); 5) \(\frac{5 + 3x - 4x^2}{2x^2}\); 6) \(\frac{6y - 2xy - 3}{3y^2}\); 7) \(\frac{8b - 20 - 3ab - 3a}{24a}\); 8) \(\frac{3ab - 3b - 4 + 10a}{42b}\); 9) \(\frac{6a - 44b - 11ab}{11b}\); 10) \(\frac{3b - 26a + 13ab}{13a}\); 11) \(\frac{9x + 2y}{12x^3y^3}\); 12) \(\frac{10x + 9y}{75x^4y^4}\); 13) \(\frac{11 - 7a}{15(a - 1)}\); 14) \(\frac{4b + 11}{30(b - 2)}\); 15) \(\frac{1}{6(m - n)}\); 16) \(\frac{17}{12(p - q)}\); 17) \(\frac{2a + 1}{ab(a + b)}\); 18) \(\frac{4y - x + 2}{xy(x + y)}\); 19) \(\frac{x + 10}{9 - x^2}\); 20) \(\frac{6y + 7}{4 - y^2}\); 21) \(\frac{5a + b}{(a + b)^2}\); 22) \(\frac{5a - 6b}{(a - b)^2}\); 23) \(\frac{m + 3n}{(m + n)^2(m - n)}\); 24) \(\frac{5y - x}{(x - y)^2(x + y)}\).
