5.35. Решите уравнения:

Решение:

1. \( \frac{8x+1}{3} = \frac{5x-1}{7} \)
   \( 7(8x+1) = 3(5x-1) \)
   \( 56x + 7 = 15x - 3 \)
   \( 56x - 15x = -3 - 7 \)
   \( 41x = -10 \)
   \( x = -\frac{10}{41} \)
2. \( \frac{3x+11}{5} = \frac{5x-7}{15} \)
   \( 3(3x+11) = 5x-7 \)
   \( 9x + 33 = 5x - 7 \)
   \( 9x - 5x = -7 - 33 \)
   \( 4x = -40 \)
   \( x = -10 \)
3. \( \frac{7x-2}{111} = \frac{8x+5}{37} \)
   \( 7x-2 = 3(8x+5) \)
   \( 7x-2 = 24x+15 \)
   \( 7x-24x = 15+2 \)
   \( -17x = 17 \)
   \( x = -1 \)
4. \( \frac{x+3}{6} + \frac{x-7}{3} = \frac{x+11}{2} \)
   \( x+3 + 2(x-7) = 3(x+11) \)
   \( x+3 + 2x-14 = 3x+33 \)
   \( 3x-11 = 3x+33 \)
   \( -11 = 33 \) — это неверно, решений нет.
5. \( x + \frac{3x-9}{5} = 11 - \frac{15x-12}{3} \)
   \( x + \frac{3x-9}{5} = 11 - (5x-4) \)
   \( x + \frac{3x-9}{5} = 11 - 5x + 4 \)
   \( x + \frac{3x-9}{5} = 15 - 5x \)
   \( \frac{5x+3x-9}{5} = 15 - 5x \)
   \( 8x-9 = 5(15-5x) \)
   \( 8x-9 = 75-25x \)
   \( 8x+25x = 75+9 \)
   \( 33x = 84 \)
   \( x = \frac{84}{33} = \frac{28}{11} \)
6. \( \frac{8x-3}{11} - \frac{5x+2}{2} = 1 \)
   \( 2(8x-3) - 11(5x+2) = 22 \)
   \( 16x-6 - 55x-22 = 22 \)
   \( -39x - 28 = 22 \)
   \( -39x = 22+28 \)
   \( -39x = 50 \)
   \( x = -\frac{50}{39} \)
7. \( \frac{3x+1}{5} - \frac{2x-1}{3} = \frac{7x+3}{15} \)
   \( 3(3x+1) - 5(2x-1) = 7x+3 \)
   \( 9x+3 - 10x+5 = 7x+3 \)
   \( -x+8 = 7x+3 \)
   \( 8-3 = 7x+x \)
   \( 5 = 8x \)
   \( x = \frac{5}{8} \)
8. \( \frac{x+3}{6} + \frac{x-7}{3} = \frac{x+11}{2} \)
   \( x+3 + 2(x-7) = 3(x+11) \)
   \( x+3+2x-14 = 3x+33 \)
   \( 3x-11 = 3x+33 \)
   \( -11 = 33 \) — это неверно, решений нет.
9. \( \frac{x-1}{7} + \frac{23-x}{5} = 2 - \frac{4+x}{4} \)
   \( 20(x-1) + 28(23-x) = 40 - 35(4+x) \)
   \( 20x-20 + 644-28x = 40 - 140-35x \)
   \( -8x + 624 = -100 - 35x \)
   \( -8x + 35x = -100 - 624 \)
   \( 27x = -724 \)
   \( x = -\frac{724}{27} \)
10. \( 1 - (1-(1-(1-(1-x)))) = 57 \)
    \( 1 - (1-(1-(1-1+x))) = 57 \)
    \( 1 - (1-(1-(x))) = 57 \)
    \( 1 - (1-1+x) = 57 \)
    \( 1 - x = 57 \)
    \( x = 1 - 57 \)
    \( x = -56 \)
11. \( 1 - 2(1 - 3(1 - 4(1 - 5(1 - x)))) = 57 \)
    \( 1 - 2(1 - 3(1 - 4(1 - 5 + 5x))) = 57 \)
    \( 1 - 2(1 - 3(1 - 4(-4 + 5x))) = 57 \)
    \( 1 - 2(1 - 3(1 + 16 - 20x)) = 57 \)
    \( 1 - 2(1 - 3(17 - 20x)) = 57 \)
    \( 1 - 2(1 - 51 + 60x) = 57 \)
    \( 1 - 2(-50 + 60x) = 57 \)
    \( 1 + 100 - 120x = 57 \)
    \( 101 - 120x = 57 \)
    \( 101 - 57 = 120x \)
    \( 44 = 120x \)
    \( x = \frac{44}{120} = \frac{11}{30} \)

Ответ: 1) x = -10/41; 2) x = -10; 3) x = -1; 4) решений нет; 5) x = 28/11; 6) x = -50/39; 7) x = 5/8; 8) решений нет; 9) x = -724/27; 10) x = -56; 11) x = 11/30.