Finite Mathematics, Chapter 1, Review Exercises, Section Review Exercises, Problem 36

Determine the equilibrium quantity and price for the diet pills if the supply and demand function
is $\displaystyle S(q) = \frac{1}{2} q + 10$ and $\displaystyle D(q) = -\frac{1}{2}q + 72.50$ respectively.

The equilibrium quantity is found when the prices from both supply and demand are equal. This is $D(q) = S(q)$

$
\begin{equation}
\begin{aligned}
-\frac{1}{2} q+ 72.50 &= \frac{1}{2} q + 10\\
\\
-\frac{1}{2} q- \frac{1}{2} q &= 10 - 72.50\\
\\
-q &= -62.50\\
\\
q &= 62.50
\end{aligned}
\end{equation}
$


Therefore, the equilibrium price is

$
\begin{equation}
\begin{aligned}
S(62.50) &= \frac{1}{2}(62.50) + 10 \\
\\
&= 41.25
\end{aligned}
\end{equation}
$


In other words, there is no loss or profit if the number of pills produced is $62.50$