$\frac{2y}{y - 3} = \frac{3y + 3}{y}$
y ≠ 0
и
y − 3 ≠ 0
y ≠ 3
$\frac{2y}{y - 3} = \frac{3y + 3}{y}$ | * y(y − 3)
2y * y = (3y + 3)(y − 3)
$2y^2 = 3y^2 + 3y - 9y - 9$
$2y^2 - 3y^2 + 6y + 9 = 0$
$-y^2 + 6y + 9 = 0$
$D = b^2 - 4ac = 6^2 - 4 * (-1) * 9 = 36 + 36 = 72 > 0$
$y_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-6 + \sqrt{72}}{2 * (-1)} = \frac{-6 + 6\sqrt{2}}{-2} = 3 - 3\sqrt{2}$
$y_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-6 - \sqrt{72}}{2 * (-1)} = \frac{-6 - 6\sqrt{2}}{-2} = 3 + 3\sqrt{2}$
Ответ: $3 - 3\sqrt{2}$ и $3 + 3\sqrt{2}$

$\frac{3x + 4}{x - 3} = \frac{2x - 9}{x + 1}$
x − 3 ≠ 0
x ≠ 3
и
x + 1 ≠ 0
x ≠ −1
$\frac{3x + 4}{x - 3} - \frac{2x - 9}{x + 1} = 0$ | * (x − 3)(x + 1)
(3x + 4)(x + 1) − (2x − 9)(x − 3) = 0
$3x^2 + 4x + 3x + 4 - (2x^2 - 9x - 6x + 27) = 0$
$3x^2 + 7x + 4 - 2x^2 + 15x - 27 = 0$
$x^2 + 22x - 23 = 0$
$D = b^2 - 4ac = 22^2 - 4 * 1 * (-23) = 484 + 92 = 576 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-22 + \sqrt{576}}{2 * 1} = \frac{-22 + 24}{2} = \frac{2}{2} = 1$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-22 - \sqrt{576}}{2 * 1} = \frac{-22 - 24}{2} = \frac{-46}{2} = -23$
Ответ: −23 и 1

$\frac{5x + 2}{x - 1} = \frac{4x + 13}{x + 7}$
x − 1 ≠ 0
x ≠ 1
и
x + 7 ≠ 0
x ≠ −7
$\frac{5x + 2}{x - 1} - \frac{4x + 13}{x + 7} = 0$ | * (x − 1)(x + 7)
(5x + 2)(x + 7) − (4x + 13)(x − 1) = 0
$5x^2 + 2x + 35x + 14 - (4x^2 + 13x - 4x - 13) = 0$
$5x^2 + 37x + 14 - 4x^2 - 9x + 13 = 0$
$x^2 + 28x + 27 = 0$
$D = b^2 - 4ac = 28^2 - 4 * 1 * 27 = 784 - 108 = 676 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-28 + \sqrt{676}}{2 * 1} = \frac{-28 + 26}{2} = \frac{-2}{2} = -1$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-28 - \sqrt{676}}{2 * 1} = \frac{-28 - 26}{2} = \frac{-54}{2} = -27$
Ответ: −27 и −1

$\frac{2x^2 - 3x + 1}{x - 1} = 3x - 4$
x − 1 ≠ 0
x ≠ 1
$\frac{2x^2 - 3x + 1}{x - 1} - (3x - 4) = 0$ | * (x − 1)
$2x^2 - 3x + 1 - (3x - 4)(x - 1) = 0$
$2x^2 - 3x + 1 - (3x^2 - 4x - 3x + 4) = 0$
$2x^2 - 3x + 1 - 3x^2 + 7x - 4 = 0$
$-x^2 + 4x - 3 = 0$ | * (−1)
$x^2 - 4x + 3 = 0$
$D = b^2 - 4ac = (-4)^2 - 4 * 1 * 3 = 16 - 12 = 4 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{4 + \sqrt{4}}{2 * 1} = \frac{4 + 2}{2} = \frac{6}{2} = 3$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{4 - \sqrt{4}}{2 * 1} = \frac{4 - 2}{2} = \frac{2}{2} = 1$ − не удовлетворяет условию, так как x ≠ 1.
Ответ: 3