4. Решите уравнение: 1) $$x+\frac{4}{15}x=6\frac{1}{3}$$; 2) $$\frac{3}{16}x+\frac{1}{4}=5\frac{7}{8}$$; 3) $$\frac{1}{2}x+\frac{1}{3}x+\frac{1}{4}x=\frac{26}{27}$$; 4) $$2\frac{1}{3}x-2\frac{3}{5}=1\frac{2}{15}$$; 5) $$4\frac{5}{12}-5\frac{3}{5}x=2\frac{2}{3}$$; 6) $$\frac{5}{18}x+2\frac{1}{2}=10\frac{5}{6}$$.

Решение:

1. $$x + \frac{4}{15}x = 6\frac{1}{3}$$ $$x(1 + \frac{4}{15}) = \frac{19}{3}$$ $$x(\frac{15}{15} + \frac{4}{15}) = \frac{19}{3}$$ $$x \cdot \frac{19}{15} = \frac{19}{3}$$ $$x = \frac{19}{3} : \frac{19}{15}$$ $$x = \frac{19}{3} \cdot \frac{15}{19}$$ $$x = \frac{19 \cdot 3 \cdot 5}{3 \cdot 19}$$ $$x = 5$$

   Ответ: 5

2. $$\frac{3}{16}x + \frac{1}{4} = 5\frac{7}{8}$$ $$\frac{3}{16}x = \frac{47}{8} - \frac{1}{4}$$ $$\frac{3}{16}x = \frac{47}{8} - \frac{2}{8}$$ $$\frac{3}{16}x = \frac{45}{8}$$ $$x = \frac{45}{8} : \frac{3}{16}$$ $$x = \frac{45}{8} \cdot \frac{16}{3}$$ $$x = \frac{15 \cdot 3 \cdot 2 \cdot 8}{8 \cdot 3}$$ $$x = 15 \cdot 2$$ $$x = 30$$

   Ответ: 30

3. $$\frac{1}{2}x + \frac{1}{3}x + \frac{1}{4}x = \frac{26}{27}$$ $$x(\frac{1}{2} + \frac{1}{3} + \frac{1}{4}) = \frac{26}{27}$$ $$x(\frac{6}{12} + \frac{4}{12} + \frac{3}{12}) = \frac{26}{27}$$ $$x \cdot \frac{13}{12} = \frac{26}{27}$$ $$x = \frac{26}{27} : \frac{13}{12}$$ $$x = \frac{26}{27} \cdot \frac{12}{13}$$ $$x = \frac{2 \cdot 13 \cdot 4 \cdot 3}{9 \cdot 3 \cdot 13}$$ $$x = \frac{2 \cdot 4}{9}$$ $$x = \frac{8}{9}$$

   Ответ: $$\frac{8}{9}$$

4. $$2\frac{1}{3}x - 2\frac{3}{5} = 1\frac{2}{15}$$ $$\frac{7}{3}x - \frac{13}{5} = \frac{17}{15}$$ $$\frac{7}{3}x = \frac{17}{15} + \frac{13}{5}$$ $$\frac{7}{3}x = \frac{17}{15} + \frac{39}{15}$$ $$\frac{7}{3}x = \frac{56}{15}$$ $$x = \frac{56}{15} : \frac{7}{3}$$ $$x = \frac{56}{15} \cdot \frac{3}{7}$$ $$x = \frac{8 \cdot 7 \cdot 3}{5 \cdot 3 \cdot 7}$$ $$x = \frac{8}{5} = 1\frac{3}{5}$$

   Ответ: $$1\frac{3}{5}$$

5. $$4\frac{5}{12} - 5\frac{3}{5}x = 2\frac{2}{3}$$ $$\frac{53}{12} - \frac{28}{5}x = \frac{8}{3}$$ $$-\frac{28}{5}x = \frac{8}{3} - \frac{53}{12}$$ $$-\frac{28}{5}x = \frac{32}{12} - \frac{53}{12}$$ $$-\frac{28}{5}x = -\frac{21}{12}$$ $$-\frac{28}{5}x = -\frac{7}{4}$$ $$x = -\frac{7}{4} : (-\frac{28}{5})$$ $$x = \frac{7}{4} \cdot \frac{5}{28}$$ $$x = \frac{7 \cdot 5}{4 \cdot 4 \cdot 7}$$ $$x = \frac{5}{16}$$

   Ответ: $$\frac{5}{16}$$

6. $$\frac{5}{18}x + 2\frac{1}{2} = 10\frac{5}{6}$$ $$\frac{5}{18}x + \frac{5}{2} = \frac{65}{6}$$ $$\frac{5}{18}x = \frac{65}{6} - \frac{5}{2}$$ $$\frac{5}{18}x = \frac{65}{6} - \frac{15}{6}$$ $$\frac{5}{18}x = \frac{50}{6}$$ $$\frac{5}{18}x = \frac{25}{3}$$ $$x = \frac{25}{3} : \frac{5}{18}$$ $$x = \frac{25}{3} \cdot \frac{18}{5}$$ $$x = \frac{5 \cdot 5 \cdot 6 \cdot 3}{3 \cdot 5}$$ $$x = 5 \cdot 6$$ $$x = 30$$

   Ответ: 30