ax+8y=20 (уравнение)

Найду корень уравнения: ax+8y=20

Вы ввели

a*x + 8*y = 20

$$a x + 8 y = 20$$

Подробное решение

Дано линейное уравнение:

a*x+8*y = 20

Разделим обе части ур-ния на (8*y + a*x)/x

x = 20 / ((8*y + a*x)/x)

Получим ответ: x = 4*(5 - 2*y)/a

Быстрый ответ

       /   8*im(y)*re(a)    4*(5 - 2*re(y))*im(a)\    8*im(a)*im(y)    4*(5 - 2*re(y))*re(a)
x1 = I*|- --------------- - ---------------------| - --------------- + ---------------------
       |    2        2           2        2      |     2        2           2        2      
       \  im (a) + re (a)      im (a) + re (a)   /   im (a) + re (a)      im (a) + re (a)   

$$x_{1} = \frac{4 \left(5 - 2 \operatorname{re}{\left(y\right)}\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(- \frac{4 \left(5 - 2 \operatorname{re}{\left(y\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{8 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) - \frac{8 \operatorname{im}{\left(a\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$

Сумма и произведение корней

сумма

  /   8*im(y)*re(a)    4*(5 - 2*re(y))*im(a)\    8*im(a)*im(y)    4*(5 - 2*re(y))*re(a)
I*|- --------------- - ---------------------| - --------------- + ---------------------
  |    2        2           2        2      |     2        2           2        2      
  \  im (a) + re (a)      im (a) + re (a)   /   im (a) + re (a)      im (a) + re (a)   

$$\frac{4 \left(5 - 2 \operatorname{re}{\left(y\right)}\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(- \frac{4 \left(5 - 2 \operatorname{re}{\left(y\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{8 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) - \frac{8 \operatorname{im}{\left(a\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$

=

  /   8*im(y)*re(a)    4*(5 - 2*re(y))*im(a)\    8*im(a)*im(y)    4*(5 - 2*re(y))*re(a)
I*|- --------------- - ---------------------| - --------------- + ---------------------
  |    2        2           2        2      |     2        2           2        2      
  \  im (a) + re (a)      im (a) + re (a)   /   im (a) + re (a)      im (a) + re (a)   

$$\frac{4 \left(5 - 2 \operatorname{re}{\left(y\right)}\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(- \frac{4 \left(5 - 2 \operatorname{re}{\left(y\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{8 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) - \frac{8 \operatorname{im}{\left(a\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$

произведение

  /   8*im(y)*re(a)    4*(5 - 2*re(y))*im(a)\    8*im(a)*im(y)    4*(5 - 2*re(y))*re(a)
I*|- --------------- - ---------------------| - --------------- + ---------------------
  |    2        2           2        2      |     2        2           2        2      
  \  im (a) + re (a)      im (a) + re (a)   /   im (a) + re (a)      im (a) + re (a)   

$$\frac{4 \left(5 - 2 \operatorname{re}{\left(y\right)}\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(- \frac{4 \left(5 - 2 \operatorname{re}{\left(y\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{8 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) - \frac{8 \operatorname{im}{\left(a\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$

=

4*(I*((-5 + 2*re(y))*im(a) - 2*im(y)*re(a)) - (-5 + 2*re(y))*re(a) - 2*im(a)*im(y))
-----------------------------------------------------------------------------------
                                    2        2                                     
                                  im (a) + re (a)                                  

$$\frac{4 \left(i \left(\left(2 \operatorname{re}{\left(y\right)} - 5\right) \operatorname{im}{\left(a\right)} - 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(y\right)}\right) - \left(2 \operatorname{re}{\left(y\right)} - 5\right) \operatorname{re}{\left(a\right)} - 2 \operatorname{im}{\left(a\right)} \operatorname{im}{\left(y\right)}\right)}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$

Решение параметрического уравнения

Дано уравнение с параметром:
$$a x + 8 y = 20$$
Коэффициент при x равен
$$a$$
тогда возможные случаи для a :
$$a < 0$$
$$a = 0$$
Рассмотри все случаи подробнее:
При
$$a < 0$$
уравнение будет
$$- x + 8 y - 20 = 0$$
его решение
$$x = 8 y - 20$$
При
$$a = 0$$
уравнение будет
$$8 y - 20 = 0$$
его решение