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tgx=3x+5y la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
tan(x) = 3*x + 5*y
tan⁡(x)=3x+5y\tan{\left(x \right)} = 3 x + 5 y
Simplificar
Solución detallada
Tenemos la ecuación
tan⁡(x)=3x+5y\tan{\left(x \right)} = 3 x + 5 y
cambiamos
−3x−5y+tan⁡(x)−1=0- 3 x - 5 y + \tan{\left(x \right)} - 1 = 0
−3x−5y+tan⁡(x)−1=0- 3 x - 5 y + \tan{\left(x \right)} - 1 = 0
Sustituimos
w=tan⁡(x)w = \tan{\left(x \right)}
Sumamos los términos semejantes en el miembro izquierdo de la ecuación:
-1 + w - 5*y - 3*x = 0

Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
w−3x−5y=1w - 3 x - 5 y = 1
Move the summands with the other variables
del miembro izquierdo al derecho, obtenemos:
w+(−3)x=5y+1w + \left(-3\right) x = 5 y + 1
Dividamos ambos miembros de la ecuación en (w - 3*x)/w
w = 1 + 5*y / ((w - 3*x)/w)

Obtenemos la respuesta: w = 1 + 3*x + 5*y
hacemos cambio inverso
tan⁡(x)=w\tan{\left(x \right)} = w
sustituimos w:
Gráfica
Respuesta rápida [src] 
       3*re(x)     /  3*im(x)            sinh(2*im(x))          \             sin(2*re(x))          
y1 = - ------- + I*|- ------- + --------------------------------| + --------------------------------
          5        \     5      5*(cos(2*re(x)) + cosh(2*im(x)))/   5*(cos(2*re(x)) + cosh(2*im(x)))
y1=i(−3im⁡(x)5+sinh⁡(2im⁡(x))5(cos⁡(2re⁡(x))+cosh⁡(2im⁡(x))))−3re⁡(x)5+sin⁡(2re⁡(x))5(cos⁡(2re⁡(x))+cosh⁡(2im⁡(x)))y_{1} = i \left(- \frac{3 \operatorname{im}{\left(x\right)}}{5} + \frac{\sinh{\left(2 \operatorname{im}{\left(x\right)} \right)}}{5 \left(\cos{\left(2 \operatorname{re}{\left(x\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)}\right)}\right) - \frac{3 \operatorname{re}{\left(x\right)}}{5} + \frac{\sin{\left(2 \operatorname{re}{\left(x\right)} \right)}}{5 \left(\cos{\left(2 \operatorname{re}{\left(x\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)}\right)} 
y1 = i*(-3*im(x)/5 + sinh(2*im(x))/(5*(cos(2*re(x)) + cosh(2*im(x))))) - 3*re(x)/5 + sin(2*re(x))/(5*(cos(2*re(x)) + cosh(2*im(x))))
Suma y producto de raíces [src] 
suma
  3*re(x)     /  3*im(x)            sinh(2*im(x))          \             sin(2*re(x))          
- ------- + I*|- ------- + --------------------------------| + --------------------------------
     5        \     5      5*(cos(2*re(x)) + cosh(2*im(x)))/   5*(cos(2*re(x)) + cosh(2*im(x)))
i(−3im⁡(x)5+sinh⁡(2im⁡(x))5(cos⁡(2re⁡(x))+cosh⁡(2im⁡(x))))−3re⁡(x)5+sin⁡(2re⁡(x))5(cos⁡(2re⁡(x))+cosh⁡(2im⁡(x)))i \left(- \frac{3 \operatorname{im}{\left(x\right)}}{5} + \frac{\sinh{\left(2 \operatorname{im}{\left(x\right)} \right)}}{5 \left(\cos{\left(2 \operatorname{re}{\left(x\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)}\right)}\right) - \frac{3 \operatorname{re}{\left(x\right)}}{5} + \frac{\sin{\left(2 \operatorname{re}{\left(x\right)} \right)}}{5 \left(\cos{\left(2 \operatorname{re}{\left(x\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)}\right)} 
=
  3*re(x)     /  3*im(x)            sinh(2*im(x))          \             sin(2*re(x))          
- ------- + I*|- ------- + --------------------------------| + --------------------------------
     5        \     5      5*(cos(2*re(x)) + cosh(2*im(x)))/   5*(cos(2*re(x)) + cosh(2*im(x)))
i(−3im⁡(x)5+sinh⁡(2im⁡(x))5(cos⁡(2re⁡(x))+cosh⁡(2im⁡(x))))−3re⁡(x)5+sin⁡(2re⁡(x))5(cos⁡(2re⁡(x))+cosh⁡(2im⁡(x)))i \left(- \frac{3 \operatorname{im}{\left(x\right)}}{5} + \frac{\sinh{\left(2 \operatorname{im}{\left(x\right)} \right)}}{5 \left(\cos{\left(2 \operatorname{re}{\left(x\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)}\right)}\right) - \frac{3 \operatorname{re}{\left(x\right)}}{5} + \frac{\sin{\left(2 \operatorname{re}{\left(x\right)} \right)}}{5 \left(\cos{\left(2 \operatorname{re}{\left(x\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)}\right)} 
producto
  3*re(x)     /  3*im(x)            sinh(2*im(x))          \             sin(2*re(x))          
- ------- + I*|- ------- + --------------------------------| + --------------------------------
     5        \     5      5*(cos(2*re(x)) + cosh(2*im(x)))/   5*(cos(2*re(x)) + cosh(2*im(x)))
i(−3im⁡(x)5+sinh⁡(2im⁡(x))5(cos⁡(2re⁡(x))+cosh⁡(2im⁡(x))))−3re⁡(x)5+sin⁡(2re⁡(x))5(cos⁡(2re⁡(x))+cosh⁡(2im⁡(x)))i \left(- \frac{3 \operatorname{im}{\left(x\right)}}{5} + \frac{\sinh{\left(2 \operatorname{im}{\left(x\right)} \right)}}{5 \left(\cos{\left(2 \operatorname{re}{\left(x\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)}\right)}\right) - \frac{3 \operatorname{re}{\left(x\right)}}{5} + \frac{\sin{\left(2 \operatorname{re}{\left(x\right)} \right)}}{5 \left(\cos{\left(2 \operatorname{re}{\left(x\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)}\right)} 
=
-(-sin(2*re(x)) + I*(-sinh(2*im(x)) + 3*(cos(2*re(x)) + cosh(2*im(x)))*im(x)) + 3*(cos(2*re(x)) + cosh(2*im(x)))*re(x)) 
------------------------------------------------------------------------------------------------------------------------
                                            5*cos(2*re(x)) + 5*cosh(2*im(x))
−i(3(cos⁡(2re⁡(x))+cosh⁡(2im⁡(x)))im⁡(x)−sinh⁡(2im⁡(x)))+3(cos⁡(2re⁡(x))+cosh⁡(2im⁡(x)))re⁡(x)−sin⁡(2re⁡(x))5cos⁡(2re⁡(x))+5cosh⁡(2im⁡(x))- \frac{i \left(3 \left(\cos{\left(2 \operatorname{re}{\left(x\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)}\right) \operatorname{im}{\left(x\right)} - \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)}\right) + 3 \left(\cos{\left(2 \operatorname{re}{\left(x\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)}\right) \operatorname{re}{\left(x\right)} - \sin{\left(2 \operatorname{re}{\left(x\right)} \right)}}{5 \cos{\left(2 \operatorname{re}{\left(x\right)} \right)} + 5 \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)}} 
-(-sin(2*re(x)) + i*(-sinh(2*im(x)) + 3*(cos(2*re(x)) + cosh(2*im(x)))*im(x)) + 3*(cos(2*re(x)) + cosh(2*im(x)))*re(x))/(5*cos(2*re(x)) + 5*cosh(2*im(x)))
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