Solución detallada
Tenemos una ecuación lineal:
3*a*x-3 = 3*x-a
Sumamos los términos semejantes en el miembro derecho de la ecuación:
-3 + 3*a*x = -a + 3*x
Transportamos los términos libres (sin x)
del miembro izquierdo al derecho, obtenemos:
$$3 a x = - a + 3 x + 3$$
Dividamos ambos miembros de la ecuación en 3*a
x = 3 - a + 3*x / (3*a)
Obtenemos la respuesta: x = (3 - a)/(3*(-1 + a))
Resolución de la ecuación paramétrica
Se da la ecuación con parámetro:
$$3 a x - 3 = - a + 3 x$$
Коэффициент при x равен
$$3 a - 3$$
entonces son posibles los casos para a :
$$a < 1$$
$$a = 1$$
Consideremos todos los casos con detalles:
Con
$$a < 1$$
la ecuación será
$$- 3 x - 3 = 0$$
su solución
$$x = -1$$
Con
$$a = 1$$
la ecuación será
$$-2 = 0$$
su solución
no hay soluciones
2
/ (-1 + re(a))*im(a) (3 - re(a))*im(a) \ im (a) (-1 + re(a))*(3 - re(a))
x1 = I*|- -------------------------- - --------------------------| - -------------------------- + --------------------------
| / 2 2 \ / 2 2 \| / 2 2 \ / 2 2 \
\ 3*\(-1 + re(a)) + im (a)/ 3*\(-1 + re(a)) + im (a)// 3*\(-1 + re(a)) + im (a)/ 3*\(-1 + re(a)) + im (a)/
$$x_{1} = \frac{\left(3 - \operatorname{re}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} - 1\right)}{3 \left(\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(- \frac{\left(3 - \operatorname{re}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{3 \left(\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(\operatorname{re}{\left(a\right)} - 1\right) \operatorname{im}{\left(a\right)}}{3 \left(\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{3 \left(\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}$$
x1 = (3 - re(a))*(re(a) - 1)/(3*((re(a) - 1)^2 + im(a)^2)) + i*(-(3 - re(a))*im(a)/(3*((re(a) - 1)^2 + im(a)^2)) - (re(a) - 1)*im(a)/(3*((re(a) - 1)^2 + im(a)^2))) - im(a)^2/(3*((re(a) - 1)^2 + im(a)^2))
Suma y producto de raíces
[src]
2
/ (-1 + re(a))*im(a) (3 - re(a))*im(a) \ im (a) (-1 + re(a))*(3 - re(a))
I*|- -------------------------- - --------------------------| - -------------------------- + --------------------------
| / 2 2 \ / 2 2 \| / 2 2 \ / 2 2 \
\ 3*\(-1 + re(a)) + im (a)/ 3*\(-1 + re(a)) + im (a)// 3*\(-1 + re(a)) + im (a)/ 3*\(-1 + re(a)) + im (a)/
$$\frac{\left(3 - \operatorname{re}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} - 1\right)}{3 \left(\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(- \frac{\left(3 - \operatorname{re}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{3 \left(\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(\operatorname{re}{\left(a\right)} - 1\right) \operatorname{im}{\left(a\right)}}{3 \left(\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{3 \left(\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}$$
2
/ (-1 + re(a))*im(a) (3 - re(a))*im(a) \ im (a) (-1 + re(a))*(3 - re(a))
I*|- -------------------------- - --------------------------| - -------------------------- + --------------------------
| / 2 2 \ / 2 2 \| / 2 2 \ / 2 2 \
\ 3*\(-1 + re(a)) + im (a)/ 3*\(-1 + re(a)) + im (a)// 3*\(-1 + re(a)) + im (a)/ 3*\(-1 + re(a)) + im (a)/
$$\frac{\left(3 - \operatorname{re}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} - 1\right)}{3 \left(\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(- \frac{\left(3 - \operatorname{re}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{3 \left(\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(\operatorname{re}{\left(a\right)} - 1\right) \operatorname{im}{\left(a\right)}}{3 \left(\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{3 \left(\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}$$
2
/ (-1 + re(a))*im(a) (3 - re(a))*im(a) \ im (a) (-1 + re(a))*(3 - re(a))
I*|- -------------------------- - --------------------------| - -------------------------- + --------------------------
| / 2 2 \ / 2 2 \| / 2 2 \ / 2 2 \
\ 3*\(-1 + re(a)) + im (a)/ 3*\(-1 + re(a)) + im (a)// 3*\(-1 + re(a)) + im (a)/ 3*\(-1 + re(a)) + im (a)/
$$\frac{\left(3 - \operatorname{re}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} - 1\right)}{3 \left(\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(- \frac{\left(3 - \operatorname{re}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{3 \left(\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(\operatorname{re}{\left(a\right)} - 1\right) \operatorname{im}{\left(a\right)}}{3 \left(\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{3 \left(\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}$$
/ 2 \
-\im (a) + (-1 + re(a))*(-3 + re(a)) + 2*I*im(a)/
--------------------------------------------------
2 2
3*(-1 + re(a)) + 3*im (a)
$$- \frac{\left(\operatorname{re}{\left(a\right)} - 3\right) \left(\operatorname{re}{\left(a\right)} - 1\right) + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 2 i \operatorname{im}{\left(a\right)}}{3 \left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + 3 \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
-(im(a)^2 + (-1 + re(a))*(-3 + re(a)) + 2*i*im(a))/(3*(-1 + re(a))^2 + 3*im(a)^2)