$\frac{x^2}{x^2 + 1} = \frac{7x}{x^2 + 1}$
$\frac{x^2}{x^2 + 1} - \frac{7x}{x^2 + 1} = 0$
$\frac{x^2 - 7x}{x^2 + 1} = 0$
$\begin{equation*} \begin{cases} x^2 - 7x = 0 &\\ x^2 + 1 ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x(x - 7) = 0 &\\ x ∈ R & \end{cases} \end{equation*}$
Ответ: x = {0;7}

$\frac{y^2}{y^2 - 6y} = \frac{4(3 - 2y)}{y(6 - y)}$
$\frac{y^2}{y^2 - 6y} - \frac{4(3 - 2y)}{y(6 - y)} = 0$
$\frac{y^2}{y(y - 6)} + \frac{4(3 - 2y)}{y(y - 6)} = 0$
$\frac{y^2 + 4(3 - 2y)}{y(y - 6)} = 0$
$\frac{y^2 - 8y + 12}{y(y - 6)} = 0$
$\begin{equation*} \begin{cases} y^2 - 8y + 12 = 0 &\\ y(y - 6) ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} (y - 2)(y - 6) = 0 &\\ y ≠ {0;6} & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} y = {2;6} &\\ y ≠ {0;6} & \end{cases} \end{equation*}$
Ответ: y = 2

$\frac{x - 2}{x + 2} = \frac{x + 3}{x - 4}$
$\frac{x - 2}{x + 2} - \frac{x + 3}{x - 4} = 0$
$\frac{(x - 2)(x - 4) - (x + 3)(x + 2)}{(x + 2)(x - 4)} = 0$
$\frac{x^2 - 6x + 8 - (x^2 + 5x + 6)}{(x + 2)(x - 4)} = 0$
$\frac{-11x + 2}{(x + 2)(x - 4)} = 0$
$\begin{equation*} \begin{cases} -11x + 2 = 0 &\\ (x + 2)(x - 4) ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x = \frac{2}{11} &\\ x ≠ {-2;4} & \end{cases} \end{equation*}$
Ответ: $x = \frac{2}{11}$

$\frac{8y - 5}{y} = \frac{9y}{y + 2}$
$\frac{8y - 5}{y} - \frac{9y}{y + 2} = 0$
$\frac{(8y - 5)(y + 2) - y * 9y}{y(y + 2)} = 0$
$\frac{8y^2 + 11y - 10 - 9y^2}{y(y + 2)} = 0$
$\frac{-y^2 + 11y - 10}{y(y + 2)} = 0$
$\begin{equation*} \begin{cases} -y^2 + 11y - 10 = 0 &\\ y(y + 2) ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} y^2 - 11y + 10 = 0 &\\ y ≠ {-2;0} & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} (y - 1)(y - 10) = 0 &\\ y ≠ {-2;0} & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} y = {1;10} &\\ y ≠ {-2;0} & \end{cases} \end{equation*}$
Ответ: y = {1;10}

$\frac{x^2 + 3}{x^2 + 1} = 2$
$\frac{x^2 + 3}{x^2 + 1} - 2 = 0$
$\frac{x^2 + 3 - 2(x^2 + 1)}{x^2 + 1} = 0$
$\frac{-x^2 + 1}{x^2 + 1} = 0$
$\begin{equation*} \begin{cases} -x^2 + 1 = 0 &\\ x^2 + 1 ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x^2 - 1 = 0 &\\ x ∈ R & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x = ±1 &\\ x ∈ R & \end{cases} \end{equation*}$
Ответ: x = ±1

$\frac{3}{x^2 + 2} = \frac{1}{x}$
$\frac{3x - (x^2 + 2)}{x(x^2 + 2)} = 0$
$\begin{equation*} \begin{cases} -x^2 + 3x - 2 = 0 &\\ x(x^2 + 2) ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x^2 - 3x + 2 = 0 &\\ x ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} (x - 1)(x - 2) = 0 &\\ x ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x = {1;2} &\\ x ≠ 0 & \end{cases} \end{equation*}$
Ответ: x = {1;2}

$x + 2 = \frac{15}{4x + 1}$
$x + 2 - \frac{15}{4x + 1} = 0$
$\frac{(x + 2)(4x + 1) - 15}{4x + 1} = 0$
$\frac{4x^2 + 9x - 13}{4x + 1} = 0$
$\begin{equation*} \begin{cases} 4x^2 + 9x - 13 = 0 &\\ 4x + 1 ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} (4x + 13)(x - 1) = 0 &\\ x ≠ -0,25 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x = {-3,25;1} &\\ x ≠ -0,25 & \end{cases} \end{equation*}$
Ответ: x = {−3,25;1}

$\frac{x^2 - 5}{x - 1} = \frac{7x + 10}{9}$
$\frac{x^2 - 5}{x - 1} - \frac{7x + 10}{9} = 0$
$\frac{9(x^2 - 5) - (7x + 10)(x - 1)}{9(x - 1)}$
$\frac{2x^2 - 3x - 35}{9(x - 1)} = 0$
$\begin{equation*} \begin{cases} 2x^2 - 3x - 35 = 0 &\\ 9(x - 1) ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} (2x + 7)(x - 5) = 0 &\\ x ≠ 1 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x = {-3,5;5} &\\ x ≠ 1 & \end{cases} \end{equation*}$
Ответ: x = {−3,5;5}