Решение: Сложение и вычитание дробей с разными знаменателями

С-7. Сложение и вычитание дробей с разными знаменателями 1. Выполните сложение или вычитание дробей: 1) а) \(\frac{x}{3} + \frac{x-2}{5} = \frac{5x + 3(x-2)}{15} = \frac{5x + 3x - 6}{15} = \frac{8x - 6}{15}\) б) \(\frac{3y-2}{6} - \frac{y+1}{4} = \frac{2(3y-2) - 3(y+1)}{12} = \frac{6y - 4 - 3y - 3}{12} = \frac{3y - 7}{12}\) в) \(-\frac{b-c}{7} + \frac{3b-c}{14} = \frac{-2(b-c) + (3b-c)}{14} = \frac{-2b + 2c + 3b - c}{14} = \frac{b + c}{14}\) г) \(\frac{1}{a^2} + \frac{a-2}{a} = \frac{1 + a(a-2)}{a^2} = \frac{1 + a^2 - 2a}{a^2} = \frac{(a-1)^2}{a^2}\) д) \(\frac{3x-5}{x} - \frac{3y-2}{y} = \frac{y(3x-5) - x(3y-2)}{xy} = \frac{3xy - 5y - 3xy + 2x}{xy} = \frac{2x - 5y}{xy}\) е) \(\frac{b-a}{ab} - \frac{a-b}{b^2} = \frac{b(b-a) - a(a-b)}{ab^2} = \frac{b^2 - ab - a^2 + ab}{ab^2} = \frac{b^2 - a^2}{ab^2}\) 2) а) \(\frac{(x+y)^2}{6y} + \frac{(x-y)^2}{12y} - \frac{x^2-y^2}{4y} = \frac{2(x+y)^2 + (x-y)^2 - 3(x^2-y^2)}{12y} = \frac{2(x^2+2xy+y^2) + (x^2-2xy+y^2) - 3x^2 + 3y^2}{12y} = \frac{2x^2+4xy+2y^2+x^2-2xy+y^2-3x^2+3y^2}{12y} = \frac{2xy+6y^2}{12y} = \frac{2y(x+3y)}{12y} = \frac{x+3y}{6}\) б) \(\frac{3a+1}{7a} - \frac{7a+b}{14ab} - \frac{b-1}{2b} = \frac{2b(3a+1) - (7a+b) - 7a(b-1)}{14ab} = \frac{6ab+2b-7a-b-7ab+7a}{14ab} = \frac{b-ab}{14ab} = \frac{b(1-a)}{14ab} = \frac{1-a}{14a}\) 3) а) \(\frac{a-1}{2(a-4)} + \frac{a}{a-4} = \frac{a-1 + 2a}{2(a-4)} = \frac{3a-1}{2(a-4)}\) б) \(\frac{x-1}{3x-12} - \frac{x-3}{2x-8} = \frac{x-1}{3(x-4)} - \frac{x-3}{2(x-4)} = \frac{2(x-1) - 3(x-3)}{6(x-4)} = \frac{2x-2-3x+9}{6(x-4)} = \frac{7-x}{6(x-4)}\) в) \(\frac{3y}{4y-4} + \frac{2y}{5-5y} = \frac{3y}{4(y-1)} - \frac{2y}{5(y-1)} = \frac{15y - 8y}{20(y-1)} = \frac{7y}{20(y-1)}\) 4) а) \(\frac{a+1}{a^2-ab} - \frac{1-b}{b^2-ab} = \frac{a+1}{a(a-b)} - \frac{1-b}{-b(a-b)} = \frac{a+1}{a(a-b)} + \frac{1-b}{b(a-b)} = \frac{b(a+1) + a(1-b)}{ab(a-b)} = \frac{ab+b+a-ab}{ab(a-b)} = \frac{a+b}{ab(a-b)}\) 2. Представьте в виде дроби: 1) а) \(5x + \frac{1}{x} = \frac{5x^2+1}{x}\) б) \(\frac{6}{y} - 2y = \frac{6-2y^2}{y}\) в) \(4a - \frac{8a^2}{2a-3} = \frac{4a(2a-3) - 8a^2}{2a-3} = \frac{8a^2 - 12a - 8a^2}{2a-3} = \frac{-12a}{2a-3} = \frac{12a}{3-2a}\) г) \(\frac{6b}{3-b} - 2b = \frac{6b - 2b(3-b)}{3-b} = \frac{6b - 6b + 2b^2}{3-b} = \frac{2b^2}{3-b}\) 2) а) \(\frac{8b^2}{4b-5} - 2b - 1 = \frac{8b^2 - (2b+1)(4b-5)}{4b-5} = \frac{8b^2 - (8b^2 - 10b + 4b - 5)}{4b-5} = \frac{8b^2 - 8b^2 + 6b + 5}{4b-5} = \frac{6b+5}{4b-5}\) б) \(3x + \frac{3+4x-4x^2}{2x-3} + 1 = (3x+1) + \frac{3+4x-4x^2}{2x-3} = \frac{(3x+1)(2x-3) + 3+4x-4x^2}{2x-3} = \frac{6x^2 - 9x + 2x - 3 + 3 + 4x - 4x^2}{2x-3} = \frac{2x^2 - 3x}{2x-3} = \frac{x(2x-3)}{2x-3} = x\)