Решение уравнений:

1. a) $$\frac{1}{5x-9} = \frac{1}{6}$$$$\Rightarrow$$$$\ 5x-9 = 6$$ $$\Rightarrow$$$$\ 5x = 15$$ $$\Rightarrow$$$$\ x = 3$$ Ответ: $$x = 3$$
2. б) $$\frac{1}{4x-15} = \frac{1}{3x-5}$$$$\Rightarrow$$$$\ 4x-15 = 3x-5$$ $$\Rightarrow$$$$\ 4x - 3x = 15 - 5$$ $$\Rightarrow$$$$\ x = 10$$ Ответ: $$x = 10$$
3. в) $$\frac{x^2}{7-x} = \frac{5x}{7-x}$$$$\Rightarrow$$$$\ x^2 = 5x$$ $$\Rightarrow$$$$\ x^2 - 5x = 0$$ $$\Rightarrow$$$$\ x(x-5) = 0$$ $$\Rightarrow$$$$\ x_1 = 0, x_2 = 5$$ Ответ: $$x_1 = 0, x_2 = 5$$
4. г) $$\frac{x^2+18x}{x+4} = \frac{-81}{x+4}$$$$\Rightarrow$$$$\ x^2+18x = -81$$ $$\Rightarrow$$$$\ x^2+18x + 81 = 0$$ $$\Rightarrow$$$$\ (x+9)^2 = 0$$ $$\Rightarrow$$$$\ x = -9$$ Ответ: $$x = -9$$
5. д) $$\frac{2x+3}{x+2} - \frac{3x+2}{x} = 0$$$$\Rightarrow$$$$\frac{(2x+3)x - (3x+2)(x+2)}{x(x+2)} = 0$$$$\Rightarrow$$$$\frac{2x^2+3x - (3x^2+6x+2x+4)}{x(x+2)} = 0$$$$\Rightarrow$$$$\frac{2x^2+3x - 3x^2-8x-4}{x(x+2)} = 0$$$$\Rightarrow$$$$\frac{-x^2-5x-4}{x(x+2)} = 0$$$$\Rightarrow$$$$\ x^2+5x+4 = 0$$ $$\Rightarrow$$$$\ D = 5^2 - 4\cdot1\cdot4 = 25 - 16 = 9$$ $$\Rightarrow$$$$\ x_1 = \frac{-5 + 3}{2} = -1, x_2 = \frac{-5 - 3}{2} = -4$$ Ответ: $$x_1 = -1, x_2 = -4$$
6. е) $$\frac{1}{x-1} - \frac{x}{x+1} = \frac{2x+1}{x^2-1}$$$$\Rightarrow$$$$\frac{(x+1) - x(x-1)}{x^2-1} = \frac{2x+1}{x^2-1}$$$$\Rightarrow$$$$\ x+1 - x^2 + x = 2x+1$$$$\Rightarrow$$$$\ -x^2 + 2x + 1 = 2x+1$$ $$\Rightarrow$$$$\ -x^2 = 0$$ $$\Rightarrow$$$$\ x = 0$$ Ответ: $$x = 0$$