1441. Знайдіть корінь рівняння: 1) 0,6(0,5x + 2/3) = 2,25 + 5,3x; 2) 5 - y = 8y - 1/3(4,5y - 5); 3) 1/2(x - 4) + 6x = 5 - 1 1/2x; 4) 3,2(1 - 2y) = 0,7(3y - 1,5); 5) 5/12(z - 3) = 1/6(2z - 7) + 2; 6) 5/8(x - 2) = 2/3(x + 2) - (3 - x); 7) 1/6y - (0,5 + 8/9y) = 1/9y - (1/3 + y); 8) 3/5(3 - 2z) = 2/5(9 - z) - 0,3(z - 9).

Краткое пояснение:

Для решения уравнений с дробями и десятичными числами необходимо привести их к общему виду, раскрыть скобки, привести подобные слагаемые и найти значение неизвестной переменной.

Пошаговое решение:

1. 1) 0,6(0,5x + 2/3) = 2,25 + 5,3x
   \( \frac{3}{5}(\frac{1}{2}x + \frac{2}{3}) = \frac{9}{4} + \frac{53}{10}x \)
   \( \frac{3}{10}x + \frac{2}{5} = \frac{9}{4} + \frac{53}{10}x \)
   \( \frac{2}{5} - \frac{9}{4} = \frac{53}{10}x - \frac{3}{10}x \)
   \( \frac{8-45}{20} = \frac{50}{10}x \)
   \( -\frac{37}{20} = 5x \)
   \( x = -\frac{37}{20  5} = -\frac{37}{100} = -0,37 \)
2. 2) 5 - y = 8y - 1/3(4,5y - 5)
   \( 5 - y = 8y - \frac{1}{3}(\frac{9}{2}y - 5) \)
   \( 5 - y = 8y - \frac{3}{2}y + \frac{5}{3} \)
   \( 5 - \frac{5}{3} = 8y - \frac{3}{2}y + y \)
   \( \frac{15-5}{3} = (8 - \frac{3}{2} + 1)y \)
   \( \frac{10}{3} = (9 - \frac{3}{2})y \)
   \( \frac{10}{3} = \frac{18-3}{2}y \)
   \( \frac{10}{3} = \frac{15}{2}y \)
   \( y = \frac{10}{3}  \frac{2}{15} = \frac{20}{45} = \frac{4}{9} \)
3. 3) 1/2(x - 4) + 6x = 5 - 1 1/2x
   \( \frac{1}{2}x - 2 + 6x = 5 - \frac{3}{2}x \)
   \( \frac{13}{2}x - 2 = 5 - \frac{3}{2}x \)
   \( \frac{13}{2}x + \frac{3}{2}x = 5 + 2 \)
   \( \frac{16}{2}x = 7 \)
   \( 8x = 7 \)
   \( x = \frac{7}{8} \)
4. 4) 3,2(1 - 2y) = 0,7(3y - 1,5)
   \( \frac{16}{5}(1 - 2y) = \frac{7}{10}(3y - \frac{3}{2}) \)
   \( \frac{16}{5} - \frac{32}{5}y = \frac{21}{10}y - \frac{21}{20} \)
   \( \frac{16}{5} + \frac{21}{20} = \frac{21}{10}y + \frac{32}{5}y \)
   \( \frac{64+21}{20} = (\frac{21}{10} + \frac{64}{10})y \)
   \( \frac{85}{20} = \frac{85}{10}y \)
   \( \frac{17}{4} = \frac{17}{2}y \)
   \( y = \frac{17}{4}  \frac{2}{17} = \frac{2}{4} = \frac{1}{2} = 0,5 \)
5. 5) 5/12(z - 3) = 1/6(2z - 7) + 2
   \( \frac{5}{12}z - \frac{15}{12} = \frac{2}{6}z - \frac{7}{6} + 2 \)
   \( \frac{5}{12}z - \frac{5}{4} = \frac{1}{3}z - \frac{7}{6} + \frac{12}{6} \)
   \( \frac{5}{12}z - \frac{5}{4} = \frac{1}{3}z + \frac{5}{6} \)
   \( \frac{5}{12}z - \frac{1}{3}z = \frac{5}{6} + \frac{5}{4} \)
   \( (\frac{5-4}{12})z = \frac{10+15}{12} \)
   \( \frac{1}{12}z = \frac{25}{12} \)
   \( z = 25 \)
6. 6) 5/8(x - 2) = 2/3(x + 2) - (3 - x)
   \( \frac{5}{8}x - \frac{10}{8} = \frac{2}{3}x + \frac{4}{3} - 3 + x \)
   \( \frac{5}{8}x - \frac{5}{4} = \frac{5}{3}x + \frac{4}{3} - \frac{9}{3} \)
   \( \frac{5}{8}x - \frac{5}{4} = \frac{5}{3}x - \frac{5}{3} \)
   \( \frac{5}{3} - \frac{5}{4} = \frac{5}{3}x - \frac{5}{8}x \)
   \( \frac{20-15}{12} = (\frac{40-15}{24})x \)
   \( \frac{5}{12} = \frac{25}{24}x \)
   \( x = \frac{5}{12}  \frac{24}{25} = \frac{2}{5} = 0,4 \)
7. 7) 1/6y - (0,5 + 8/9y) = 1/9y - (1/3 + y)
   \( \frac{1}{6}y - \frac{1}{2} - \frac{8}{9}y = \frac{1}{9}y - \frac{1}{3} - y \)
   \( (\frac{1}{6} - \frac{8}{9})y - \frac{1}{2} = (\frac{1}{9} - 1)y - \frac{1}{3} \)
   \( (\frac{3-16}{18})y - \frac{1}{2} = (\frac{1-9}{9})y - \frac{1}{3} \)
   \( -\frac{13}{18}y - \frac{1}{2} = -\frac{8}{9}y - \frac{1}{3} \)
   \( \frac{8}{9}y - \frac{13}{18}y = \frac{1}{2} - \frac{1}{3} \)
   \( (\frac{16-13}{18})y = \frac{3-2}{6} \)
   \( \frac{3}{18}y = \frac{1}{6} \)
   \( \frac{1}{6}y = \frac{1}{6} \)
   \( y = 1 \)
8. 8) 3/5(3 - 2z) = 2/5(9 - z) - 0,3(z - 9)
   \( \frac{9}{5} - \frac{6}{5}z = \frac{18}{5} - \frac{2}{5}z - \frac{3}{10}(z - 9) \)
   \( \frac{9}{5} - \frac{6}{5}z = \frac{18}{5} - \frac{2}{5}z - \frac{3}{10}z + \frac{27}{10} \)
   \( \frac{9}{5} - \frac{18}{5} - \frac{27}{10} = - \frac{2}{5}z - \frac{3}{10}z + \frac{6}{5}z \)
   \( \frac{18-36-27}{10} = (\frac{-4-3+12}{10})z \)
   \( \frac{-45}{10} = \frac{5}{10}z \)
   \( -4,5 = 0,5z \)
   \( z = -4,5 / 0,5 = -9 \)

Ответ: 1) x = -0,37; 2) y = 4/9; 3) x = 7/8; 4) y = 0,5; 5) z = 25; 6) x = 0,4; 7) y = 1; 8) z = -9