I am trying to solve the following system of equations:
$$x = 0.09 x + 0.6p y + (1.3p - p^2) z,$$ $$y= 0.49 x + 0.16 y + 0.7 z,$$ $$z = 0.42 x + (0.84 - 0.6p) y + (p^2 - 2p + 1) z,$$ $$x + y + z = 1$$
When I attempt to use WolframAlpha to solve this, the system starts giving me values of $p$ for some reason. I just want the equations $x$, $y$, and $z$ in terms of (unknown) $p$.
I would greatly appreciate it if someone would please take the time to help me with this.
A simplistic approach
From $(1)$ extract $z$ as a function of $x$ and $y$.
Plug the result in $(2)$ and solve for $y$ as a function of $x$. These two steps give $$y=\frac{7 \left(10 p^2-13 p-13\right) x}{24 p (5 p-9)}\qquad \qquad z=\frac{7 (5 p-13) x}{20 p (5 p-9)}$$
Plug in $(3)$ to get $$\frac{49 (p-1) (5 p-13) x}{200 p (5 p-9)}=0$$ You cannot have $x=0$ since this would make $x=y=z=0$ which cannot satisfy $(4)$.
So, the only possibilities are $p=1$ or $p=\frac {13} 5$ Now, consider each case and use $(4)$ to solve.