a*x-b=1+x (уравнение)

Найду корень уравнения: a*x-b=1+x

Вы ввели

a*x - b = 1 + x

$$a x - b = x + 1$$

Подробное решение

Дано линейное уравнение:

a*x-b = 1+x

Переносим слагаемые с неизвестным x
из правой части в левую:
$$a x - b - x = 1$$
Разделим обе части ур-ния на (-b - x + a*x)/x

x = 1 / ((-b - x + a*x)/x)

Получим ответ: x = (1 + b)/(-1 + a)

Быстрый ответ

       /  (-1 + re(a))*im(b)       (1 + re(b))*im(a)   \   (1 + re(b))*(-1 + re(a))        im(a)*im(b)      
x1 = I*|---------------------- - ----------------------| + ------------------------ + ----------------------
       |            2     2                  2     2   |                2     2                   2     2   
       \(-1 + re(a))  + im (a)   (-1 + re(a))  + im (a)/    (-1 + re(a))  + im (a)    (-1 + re(a))  + im (a)

$$x_{1} = i \left(\frac{\left(\operatorname{re}{\left(a\right)} - 1\right) \operatorname{im}{\left(b\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(\operatorname{re}{\left(b\right)} + 1\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(a\right)} - 1\right) \left(\operatorname{re}{\left(b\right)} + 1\right)}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\operatorname{im}{\left(a\right)} \operatorname{im}{\left(b\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$

Сумма и произведение корней

сумма

  /  (-1 + re(a))*im(b)       (1 + re(b))*im(a)   \   (1 + re(b))*(-1 + re(a))        im(a)*im(b)      
I*|---------------------- - ----------------------| + ------------------------ + ----------------------
  |            2     2                  2     2   |                2     2                   2     2   
  \(-1 + re(a))  + im (a)   (-1 + re(a))  + im (a)/    (-1 + re(a))  + im (a)    (-1 + re(a))  + im (a)

$$i \left(\frac{\left(\operatorname{re}{\left(a\right)} - 1\right) \operatorname{im}{\left(b\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(\operatorname{re}{\left(b\right)} + 1\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(a\right)} - 1\right) \left(\operatorname{re}{\left(b\right)} + 1\right)}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\operatorname{im}{\left(a\right)} \operatorname{im}{\left(b\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$

=

  /  (-1 + re(a))*im(b)       (1 + re(b))*im(a)   \   (1 + re(b))*(-1 + re(a))        im(a)*im(b)      
I*|---------------------- - ----------------------| + ------------------------ + ----------------------
  |            2     2                  2     2   |                2     2                   2     2   
  \(-1 + re(a))  + im (a)   (-1 + re(a))  + im (a)/    (-1 + re(a))  + im (a)    (-1 + re(a))  + im (a)

$$i \left(\frac{\left(\operatorname{re}{\left(a\right)} - 1\right) \operatorname{im}{\left(b\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(\operatorname{re}{\left(b\right)} + 1\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(a\right)} - 1\right) \left(\operatorname{re}{\left(b\right)} + 1\right)}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\operatorname{im}{\left(a\right)} \operatorname{im}{\left(b\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$

произведение

  /  (-1 + re(a))*im(b)       (1 + re(b))*im(a)   \   (1 + re(b))*(-1 + re(a))        im(a)*im(b)      
I*|---------------------- - ----------------------| + ------------------------ + ----------------------
  |            2     2                  2     2   |                2     2                   2     2   
  \(-1 + re(a))  + im (a)   (-1 + re(a))  + im (a)/    (-1 + re(a))  + im (a)    (-1 + re(a))  + im (a)

$$i \left(\frac{\left(\operatorname{re}{\left(a\right)} - 1\right) \operatorname{im}{\left(b\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(\operatorname{re}{\left(b\right)} + 1\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(a\right)} - 1\right) \left(\operatorname{re}{\left(b\right)} + 1\right)}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\operatorname{im}{\left(a\right)} \operatorname{im}{\left(b\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$

=

I*((-1 + re(a))*im(b) - (1 + re(b))*im(a)) + (1 + re(b))*(-1 + re(a)) + im(a)*im(b)
-----------------------------------------------------------------------------------
                                           2     2                                 
                               (-1 + re(a))  + im (a)                              

$$\frac{i \left(\left(\operatorname{re}{\left(a\right)} - 1\right) \operatorname{im}{\left(b\right)} - \left(\operatorname{re}{\left(b\right)} + 1\right) \operatorname{im}{\left(a\right)}\right) + \left(\operatorname{re}{\left(a\right)} - 1\right) \left(\operatorname{re}{\left(b\right)} + 1\right) + \operatorname{im}{\left(a\right)} \operatorname{im}{\left(b\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$

Решение параметрического уравнения

Дано уравнение с параметром:
$$a x - b = x + 1$$
Коэффициент при x равен
$$a - 1$$
тогда возможные случаи для a :
$$a < 1$$
$$a = 1$$
Рассмотри все случаи подробнее:
При
$$a < 1$$
уравнение будет
$$- b - x - 1 = 0$$
его решение
$$x = - b - 1$$
При
$$a = 1$$
уравнение будет
$$- b - 1 = 0$$
его решение