I am trying to solve seemingly a very simple set of equations. Following two equations are the success probability for the transmit powers $P_m, P_u$.
$$x_1 = \frac{1}{1 + ab\frac{P_u}{P_m}}$$ $$x_2 = \frac{1}{1 + ac\frac{P_m}{P_u}}$$
Keeping $P_m$ constant to be -6 dB, I ran the equations in Matlab to produce the Image below. My question might be very simple, but I am still not getting the right answer. How do I get the point of intersection between the two link? (YELLOW colored point in the figure)
What I did is as follows:
$$x_1 = x_2$$ This yields
$$P_u = P_m\sqrt{\frac{b}{c}}$$
However, still, the two results do not match
If $x_1=x_2$, then
$$b\frac{P_u}{P_m}=c\frac{P_m}{P_u}$$
$$P_u^2 = \frac{c}{b}P_m^2$$
$$P_u =P_m \sqrt{\frac{c}{b}}$$