номер 124 гдз 8 класс алгебра Муравин Муравин

1)\begin{equation}\frac{1}{10x-1}+\frac{1}{5x-2}=0 \end{equation}
ОДЗ: \begin{equation}10x-1\neq 0 \end{equation} \begin{equation}10x\neq 1 \end{equation} \begin{equation}x\neq 0,1 \end{equation} \begin{equation}5x-2\neq 0 \end{equation} \begin{equation}5x\neq 2 \end{equation} \begin{equation}x\neq 0,4 \end{equation} \begin{equation}\frac{1}{10x-1}+\frac{1}{5x-2}=0 \end{equation} \begin{equation}\frac{5x-2+10x-1}{\left ( 10x-1 \right )\left ( 5x-2 \right )}=0 \end{equation}
5x - 2 + 10x - 1 = 0
15x - 3 = 0
15x = 3 \begin{equation}x=\frac{3}{15} \end{equation} \begin{equation}x=\frac{1}{5} \end{equation}
Ответ: \begin{equation}x=\frac{1}{5} \end{equation} 2)\begin{equation}\frac{3}{8-5x}+\frac{5}{2-7x}=0 \end{equation}
ОДЗ: \begin{equation}8-5x\neq 0 \end{equation} \begin{equation}5x\neq 8 \end{equation} \begin{equation}x\neq \frac{8}{5} \end{equation} \begin{equation}2-7x\neq 0 \end{equation} \begin{equation}7x\neq 2 \end{equation} \begin{equation}x\neq \frac{2}{7} \end{equation} \begin{equation}\frac{3}{8-5x}+\frac{5}{2-7x}=0 \end{equation} \begin{equation}\frac{3\cdot \left ( 2-7x \right )+5\left ( 8-5x \right )}{\left ( 8-5x \right )\left ( 2-7x \right )}=0 \end{equation}
3 · (2 - 7x) + 5(8 - 5x) = 0
6 - 21x + 40 - 25x = 0
-46x + 46 = 0
-46x = -46 \begin{equation}x=\frac{-46}{-46} \end{equation}
x = 1
Ответ: x = 1
3)\begin{equation}\frac{2x^{2}-7x+3}{2x-1}-x=1 \end{equation}
ОДЗ: \begin{equation}2x-1\neq 0 \end{equation} \begin{equation}2x\neq 1 \end{equation} \begin{equation}x\neq 0,5 \end{equation} \begin{equation}\frac{2x^{2}-7x+3}{2x-1}-x-1=0 \end{equation} \begin{equation}\frac{2x^{2}-7x+3-2x^{2}+x-2x+1}{2x-1}=0 \end{equation}
2x² - 7x + 3 - 2x² + x - 2x + 1 = 0
-8x + 4 = 0
-8x = -4 \begin{equation}x=\frac{-4}{-8} \end{equation} \begin{equation}x=\frac{1}{2} \end{equation}
x = 0,5
Ответ: решения нет
4)\begin{equation}3x-\frac{3x^{2}+2}{x+5}=4 \end{equation}
ОДЗ: \begin{equation}x+5\neq 0 \end{equation} \begin{equation}x\neq -5 \end{equation} \begin{equation}3x-\frac{3x^{2}+2}{x+5}-4=0 \end{equation} \begin{equation}\frac{3x\left ( x+5 \right )-\left ( 3x^{2}+2 \right )-4\left ( x+5 \right )}{x+5}=0 \end{equation}
3x² + 15x - 3x² - 2 - 4x - 20 = 0
11x - 22 = 0
11x = 22
x = 2
Ответ: x = 2
5)\begin{equation}\frac{1}{y}=\frac{5}{y-2}-\frac{4}{y-3} \end{equation}
ОДЗ: \begin{equation}y\neq 0 \end{equation} \begin{equation}y-2\neq 0 \end{equation} \begin{equation}y\neq 2 \end{equation} \begin{equation}y-3\neq 0 \end{equation} \begin{equation}y\neq 3 \end{equation} \begin{equation}\frac{1}{y}=\frac{5}{y-2}-\frac{4}{y-3} \end{equation}
(y - 2)(y - 3) = 5y(y - 3) - 4y(y - 2)
y² - 3y - 2y + 6 = 5y² - 15y - 4y² + 8y
-5y + 15y - 8y = -6
2y = -6
y = -3
Ответ: y = -3
6)\begin{equation}\frac{3}{z-2}+\frac{7}{z+2}=\frac{10}{z} \end{equation}
ОДЗ: \begin{equation}z-2\neq 0 \end{equation} \begin{equation}z\neq 2 \end{equation} \begin{equation}z\neq 0 \end{equation} \begin{equation}z+2\neq 0 \end{equation} \begin{equation}z\neq -2 \end{equation} \begin{equation}\frac{3}{z-2}+\frac{7}{z+2}=\frac{10}{z} \end{equation}
3z(z + 2) + 7z(z - 2) = 10 · (z² - 4)
3z² + 6z + 7z² - 14z = 10z² - 40
-8z = -40 \begin{equation}z=\frac{-40}{-8} \end{equation}
z = 5
Ответ: z = 5
7)\begin{equation}\frac{1+4x}{2}=\frac{1-4x}{1+6x}+\frac{1+6x}{3} \end{equation} \begin{equation}\frac{1+4x}{2}-\frac{1-4x}{1+6x}-\frac{1+6x}{3}=0 \end{equation}
ОДЗ: \begin{equation}1+6x\neq 0 \end{equation} \begin{equation}6x\neq -1 \end{equation} \begin{equation}x\neq -\frac{1}{6} \end{equation}
3 · (1 + 6x)(1 + 4x) - 6(1 - 4x) - 2(1 + 6x)² = 0
3 · (1 + 4x + 6x + 24x²) - 6 + 24x - 2(1 + 12x + 36x²) = 0
3 · (1 + 10x + 24x²) - 6 + 24x - 2 - 24x - 72x² = 0
3 + 30x + 72x² - 6 + 24x - 2 - 24x - 72x² = 0
30x - 5 = 0
30x = 5 \begin{equation}x=\frac{5}{30} \end{equation} \begin{equation}x=\frac{1}{6} \end{equation}
Ответ: \begin{equation}x=\frac{1}{6} \end{equation}
8)\begin{equation}\frac{2x-1}{5x-1}=\frac{2x+1}{4}-\frac{3x-1}{6} \end{equation} \begin{equation}\frac{2x-1}{5x-1}-\frac{2x+1}{4}+\frac{3x-1}{6}=0 \end{equation}
ОДЗ: \begin{equation}5x-1\neq 0 \end{equation} \begin{equation}5x\neq 1 \end{equation} \begin{equation}x\neq \frac{1}{5} \end{equation}
12(2x - 1) - 3(5x - 1)(2x + 1) + 2(3x - 1)(5x - 1) = 0
24x - 12 - 3(10x² + 5x - 2x - 1) + 2(15x² - 3x - 5x + 1) = 0
24x - 12 - 30x² - 9x + 3 + 30x² - 16x + 2 = 0
-x - 7 = 0
-x = 7
x = -7
Ответ: x = -7
9)\begin{equation}\frac{3x+9}{3x-1}+\frac{2x-13}{2x+5}=2 \end{equation} \begin{equation}\frac{3x+9}{3x-1}+\frac{2x-13}{2x+5}-2=0 \end{equation}
ОДЗ: \begin{equation}3x-1\neq 0 \end{equation} \begin{equation}3x\neq 1 \end{equation} \begin{equation}x\neq \frac{1}{3} \end{equation} \begin{equation}2x+5\neq 0 \end{equation} \begin{equation}2x\neq -5 \end{equation} \begin{equation}x\neq -2,5 \end{equation}
(2x + 5)(3x + 9) + (3x - 1)(2x - 13) - 2(2x + 5)(3x - 1) = 0
6x² + 18x + 15x + 45 + 6x² - 39x - 2x + 13 - 2(6x² - 2x + 15x - 5) = 0
12x² + 33x + 45 - 41x + 13 - 12x² + 4x - 30x + 10 = 0
-34x + 68 = 0
-34x = -68 \begin{equation}x=\frac{-68}{-34} \end{equation}
x = 2
Ответ: x = 2
10)\begin{equation}\frac{5x+13}{5x+4}+\frac{6x-4}{3x-1}=3 \end{equation} \begin{equation}\frac{5x+13}{5x+4}+\frac{6x-4}{3x-1}-3=0 \end{equation}
ОДЗ: \begin{equation}5x+4\neq 0 \end{equation} \begin{equation}5x\neq -4 \end{equation} \begin{equation}x\neq -0,8 \end{equation} \begin{equation}3x-1\neq 0 \end{equation} \begin{equation}3x\neq 1 \end{equation} \begin{equation}x\neq \frac{1}{3} \end{equation}
(3x - 1)(5x + 13) + (5x + 4)(6x - 4) - 3(3x - 1)(5x + 4) = 0
15x² + 39x - 5x - 13 + 30x² - 20x + 24x - 16 - 3(15x² + 12x - 5x - 4) = 0
45x² + 34x - 13 + 4x - 16 - 45x² - 21x + 12 = 0
17x - 17 = 0
17x = 17
x = 1
Ответ: x = 1
11)\begin{equation}\frac{y+5}{2y+6}+\frac{y+3}{3y-6}=\frac{5}{6} \end{equation} \begin{equation}\frac{y+5}{2\left ( y+3 \right )}+\frac{y+3}{3\left ( y-2 \right )}-\frac{5}{6}=0 \end{equation}
ОДЗ: \begin{equation}y+3\neq 0 \end{equation} \begin{equation}y\neq -3 \end{equation} \begin{equation}y-2\neq 0 \end{equation} \begin{equation}y\neq 2 \end{equation}
3(y - 2)(y + 5) + 2(y + 3)² - 5(y - 2)(y + 3) = 0
3(y² + 5y - 2y - 10) - 2(y² + 6y + 9) - 5(y² + 3y - 2y - 6) = 0
3y² + 9y - 30 + 2y² + 12y + 18 - 5y² - 5y + 30 = 0
16y + 18 = 0
16y = -18 \begin{equation}y=-\frac{18}{16} \end{equation} \begin{equation}y=-\frac{9}{8}=-1\frac{1}{8} \end{equation}
Ответ: \begin{equation} y=-1\frac{1}{8} \end{equation}
12)\begin{equation}\frac{y+5}{5y-20}+\frac{y-4}{3y-6}=\frac{3}{4} \end{equation}
ОДЗ: \begin{equation}5y-20\neq 0 \end{equation} \begin{equation}5y\neq 20 \end{equation} \begin{equation}y\neq 4 \end{equation} \begin{equation}3y-6\neq 0 \end{equation} \begin{equation}3y\neq 6 \end{equation} \begin{equation}y\neq 2 \end{equation} \begin{equation}\frac{y+5}{5y-20}+\frac{y-4}{3y-6}=\frac{3}{4} \end{equation}
4(3y - 6)(y + 5) + 4(5y - 20)(y - 4) = 3(5y - 20)(3y - 6)
4(3y² + 15y - 6y - 30) + 4(5y² - 20y - 20y + 80) = 3 · (15y² - 30y - 60y + 120)
12y² + 36y - 120 + 20y² - 160y + 320 = 45y² - 270y + 360
32y² - 124y + 200 - 45y² + 270y - 360 = 0
-13y² + 146y - 160 = 0