5.34. Решите уравнения:

Решение:

1. \( 0,4(x-3)=0,5(4+x)-2,5 \)
   \( 0,4x - 1,2 = 2 + 0,5x - 2,5 \)
   \( 0,4x - 0,5x = 2 - 2,5 + 1,2 \)
   \( -0,1x = 0,7 \)
   \( x = \frac{0,7}{-0,1} \)
   \( x = -7 \)
2. \( 3,96 - 1,8n = 4,7 - 2,7n \)
   \( -1,8n + 2,7n = 4,7 - 3,96 \)
   \( 0,9n = 0,74 \)
   \( n = \frac{0,74}{0,9} \)
   \( n = \frac{74}{90} = \frac{37}{45} \)
3. \( 0,2y - \frac{5}{3} = \frac{2}{5} y - 0,28 \)
   \( 0,28 + 0,2 = \frac{2}{5} y - 0,2y \)
   \( 0,48 = 0,4y - 0,2y \)
   \( 0,48 = 0,2y \)
   \( y = \frac{0,48}{0,2} \)
   \( y = 2,4 \)
4. \( 4,5 \left( \frac{7}{15} x + \frac{2}{9} \right) - 0,77 \left( \frac{8}{11} x - \frac{3}{7} \right) = -1,75 \)
   \( \frac{9}{2} \left( \frac{7}{15} x + \frac{2}{9} \right) - \frac{7}{10} \left( \frac{8}{11} x - \frac{3}{7} \right) = -1,75 \)
   \( \frac{63}{30} x + \frac{18}{18} - \frac{56}{110} x + \frac{21}{70} = -1,75 \)
   \( \frac{21}{10} x + 1 - \frac{28}{55} x + \frac{3}{10} = -1,75 \)
   \( \frac{21}{10} x - \frac{28}{55} x = -1,75 - 1 - \frac{3}{10} \)
   \( \frac{231}{110} x - \frac{56}{110} x = -2,75 - 0,3 \)
   \( \frac{175}{110} x = -3,05 \)
   \( \frac{35}{22} x = -3,05 = -\frac{305}{100} = -\frac{61}{20} \)
   \( x = -\frac{61}{20} \cdot \frac{22}{35} \)
   \( x = -\frac{61 \cdot 11}{10 \cdot 35} = -\frac{671}{350} \)
5. \( \frac{5}{7} (0,21 - 1,4x) - \frac{4}{9} (0,36 - 4,5x) = 1 \)
   \( \frac{5}{7} \left( \frac{21}{100} - \frac{14}{10} x \right) - \frac{4}{9} \left( \frac{36}{100} - \frac{45}{10} x \right) = 1 \)
   \( \frac{1}{14} - \frac{1}{5} x - \frac{16}{100} + \frac{20}{10} x = 1 \)
   \( \frac{1}{14} - \frac{1}{5} x - \frac{4}{25} + 2x = 1 \)
   \( 2x - \frac{1}{5} x = 1 - \frac{1}{14} + \frac{4}{25} \)
   \( \frac{9}{5} x = \frac{350 - 25 + 56}{350} \)
   \( \frac{9}{5} x = \frac{381}{350} \)
   \( x = \frac{381}{350} \cdot \frac{5}{9} = \frac{127}{70} \)
6. \( 3a + 5 = 8a - 15 \)
   \( 5 + 15 = 8a - 3a \)
   \( 20 = 5a \)
   \( a = 4 \)
7. \( \frac{2}{3} x + \frac{1}{3} = \frac{1}{2} x + \frac{2}{3} \)
   \( \frac{2}{3} x - \frac{1}{2} x = \frac{2}{3} - \frac{1}{3} \)
   \( \frac{4-3}{6} x = \frac{1}{3} \)
   \( \frac{1}{6} x = \frac{1}{3} \)
   \( x = 2 \)
8. \( \frac{5}{8} y - \frac{3}{4} = 2y - \frac{2}{5} \)
   \( \frac{5}{8} y - 2y = \frac{3}{4} - \frac{2}{5} \)
   \( \frac{5 - 16}{8} y = \frac{15 - 8}{20} \)
   \( -\frac{11}{8} y = \frac{7}{20} \)
   \( y = \frac{7}{20} \cdot \left( -\frac{8}{11} \right) = -\frac{7 \cdot 2}{5 \cdot 11} = -\frac{14}{55} \)
9. \( \frac{1}{2} \left( 4 - 3 \frac{1}{2} x \right) = \frac{1}{4} x - \frac{1}{2} \)
   \( \frac{1}{2} \left( 4 - \frac{7}{2} x \right) = \frac{1}{4} x - \frac{1}{2} \)
   \( 2 - \frac{7}{4} x = \frac{1}{4} x - \frac{1}{2} \)
   \( 2 + \frac{1}{2} = \frac{1}{4} x + \frac{7}{4} x \)
   \( \frac{4+1}{2} = \frac{1+7}{4} x \)
   \( \frac{5}{2} = \frac{8}{4} x \)
   \( \frac{5}{2} = 2x \)
   \( x = \frac{5}{4} \)
10. \( 2,44 + 2,3m = 3,12 + 2,7m \)
    \( 2,3m - 2,7m = 3,12 - 2,44 \)
    \( -0,4m = 0,68 \)
    \( m = \frac{0,68}{-0,4} = -1,7 \)
11. \( \frac{1}{2} y - \frac{2}{3} y = \frac{5}{9} y - \frac{7}{12} \)
    \( \frac{1}{2} y - \frac{2}{3} y - \frac{5}{9} y = -\frac{7}{12} \)
    \( \frac{9-12-10}{18} y = -\frac{7}{12} \)
    \( -\frac{13}{18} y = -\frac{7}{12} \)
    \( y = \frac{7}{12} \cdot \frac{18}{13} = \frac{7 \cdot 3}{2 \cdot 13} = \frac{21}{26} \)
12. \( 3(0,4x+7) - 4(0,8x-3) = 2 \)
    \( 1,2x + 21 - 3,2x + 12 = 2 \)
    \( -2x + 33 = 2 \)
    \( -2x = 2 - 33 \)
    \( -2x = -31 \)
    \( x = 15,5 \)
13. \( 0,4x + 0,3x - 0,84x = 1,12 \)
    \( 0,7x - 0,84x = 1,12 \)
    \( -0,14x = 1,12 \)
    \( x = \frac{1,12}{-0,14} = -8 \)
14. \( 14 + 5x = 5x + 3x \)
    \( 14 = 3x \)
    \( x = \frac{14}{3} \)
15. \( 0,7x - 1,82 = 0,8x + 3,46 \)
    \( 0,7x - 0,8x = 3,46 + 1,82 \)
    \( -0,1x = 5,28 \)
    \( x = -52,8 \)
16. \( 4,72 - 2,5x = 2x + 2,92 \)
    \( 4,72 - 2,92 = 2x + 2,5x \)
    \( 1,8 = 4,5x \)
    \( x = \frac{1,8}{4,5} = \frac{18}{45} = \frac{2}{5} = 0,4 \)
17. \( 5(x+1,2) = 12,5x \)
    \( 5x + 6 = 12,5x \)
    \( 6 = 12,5x - 5x \)
    \( 6 = 7,5x \)
    \( x = \frac{6}{7,5} = \frac{60}{75} = \frac{4}{5} = 0,8 \)
18. \( 0,2(5y-2) = 0,3(2y-1) - 0,9 \)
    \( y - 0,4 = 0,6y - 0,3 - 0,9 \)
    \( y - 0,4 = 0,6y - 1,2 \)
    \( y - 0,6y = -1,2 + 0,4 \)
    \( 0,4y = -0,8 \)
    \( y = -2 \)
19. \( 4(3-2x) + 24 = 2(3+2x) \)
    \( 12 - 8x + 24 = 6 + 4x \)
    \( 36 - 8x = 6 + 4x \)
    \( 36 - 6 = 4x + 8x \)
    \( 30 = 12x \)
    \( x = \frac{30}{12} = \frac{5}{2} = 2,5 \)

Ответ: 1) x = -7; 2) n = 37/45; 3) y = 2,4; 4) x = -671/350; 5) x = 127/70; 6) a = 4; 7) x = 2; 8) y = -14/55; 9) x = 5/4; 10) m = -1,7; 11) y = 21/26; 12) x = -8; 13) x = 14/3; 14) x = -52,8; 15) x = 0,4; 16) x = 0,8; 17) y = -2; 18) x = 2,5.