solving a system of equations (3 equations, 3 variables)

Question

I have 3 equations and 3 unknown variables as follows

$$\frac{\beta}{1-\alpha}x=y^{\alpha-1}-z$$

$$\left(1+\frac{\beta}{1-\alpha}\right)x=\frac{1}{\sigma}\left(\alpha y-\rho\right)$$

$$x\left(\frac{(1-\sigma)\beta}{1-\alpha}+\beta\frac{y^{\alpha}}{\phi z}\right)=\rho$$

Where $x,y,z$ are unknown variables and $\alpha$, $\beta$, $\rho$, $\phi$, $\sigma$ are known constant parameters.

Whatever I have tried, I could not find an analytical result for these three variables.

What possible methods could be used to solve this system ? Numerical methods or any other methods ?

Thanks in advance for hints and suggestions.

Answer 1

i will derive an equation which contains only $$y$$: from (2) we get $$x={\frac { \left( \alpha\,y-\rho \right) \left( \alpha-1 \right) }{ \sigma\, \left( \alpha-\beta-1 \right) }} $$ plugging this in (1) $${\frac { \left( \alpha\,y-\rho \right) \left( \alpha-1 \right) }{ \sigma\, \left( \alpha-\beta-1 \right) } \left( {\frac { \left( 1- \sigma \right) \beta}{1-\alpha}}+{\frac {\beta\,{y}^{\alpha}}{\Phi\,z} } \right) }=\rho$$

from (3) we obtain $$z={\frac {{y}^{\alpha-1}\alpha+x\beta-{y}^{\alpha-1}}{\alpha-1}}$$ with this equation we obatin one equation only in $$y$$ namely $${\frac { \left( \alpha\,y-\rho \right) \left( \alpha-1 \right) }{ \sigma\, \left( \alpha-\beta-1 \right) } \left( {\frac { \left( 1- \sigma \right) \beta}{1-\alpha}}+{\frac {\beta\,{y}^{\alpha} \left( \alpha-1 \right) }{\Phi\, \left( {y}^{\alpha-1}\alpha+x\beta-{y}^{ \alpha-1} \right) }} \right) }=\rho $$ where $$x={\frac { \left( \alpha\,y-\rho \right) \left( \alpha-1 \right) }{ \sigma\, \left( \alpha-\beta-1 \right) }} $$

Answer 2

In the particular case of this problem, you could notice that the equations only involve $x$ and $z$ by themselves (and not any power of them). So, the idea is to eliminate $x$ and $y$ expressing them as function of the left variable $y$ which appears with some exponents.

Since you can eliminate two variables among three, you are then left with one equation in $y$. This is what Dr. Sonnhard Graubner did.

Now, for sure, you are left with a possibly rather complex equation in $y$ which probably will not show any explicit solutions; so, numerical methods will probably be required but it is always easier to work a single nonlinear equation than the original system of equations.

Edit

Taking into account your last comment, I suspect that you tried to write a single equation in $y$. Since it is quite complex, I suggest you to define the pieces $x(y)$ and $z(y)$ and write the final equation accordingly as $$x(y)=\frac{(\alpha -1) }{\sigma (\alpha -\beta -1)}(\alpha y-\rho )$$ $$z(y)=\frac{\beta }{\alpha -1}x(y)+y^{\alpha -1}$$ $$F(y)=x(y) \left(\frac{\beta (1-\sigma )}{1-\alpha }+\frac{\beta\, y^{\alpha }}{z(y)\, \phi }\right)-\rho=0$$

Using your numbers, as I already commented, I found a single solution $$\{x= 0.219565,y= 1.572257,z=0.697140\}$$