The equilibrium point in a supply-demand system is where both supply and demand intersect. At the intersection point, x and p will be the same for both supply and demand.
We then can solve this as a system of equations. As we have both equation expressed in function of p, we can write:
[tex]\begin{gathered} p_s=p_d \\ 0.80x+58=-0.50x+162 \\ 0.80x+0.50x=162-58 \\ 1.30x=104 \\ x=\frac{104}{1.30} \\ x=80 \end{gathered}[/tex]Now that we know that x = 80 in the intersection point, we can calculate p using any of the two equations (NOTE: they should give the same result) as:
[tex]\begin{gathered} p=0.80x+58 \\ p=0.80\cdot80+58 \\ p=64+58 \\ p=122 \end{gathered}[/tex]Answer: the equilibrium point is (x,p) = (80, 122)
