Solve each problem. The supply and demand equations for a certain commodity are given. supply: p = 2000/(2000 - q) and demand: p = (7000 - 3q)/2q.
Find the equilibrium price (in dollars).

Verified step by step guidance

1. Identify the equilibrium point where supply equals demand, meaning the price from the supply equation equals the price from the demand equation. Set the two expressions for price equal to each other: \(\frac{2000}{2000 - q} = \frac{7000 - 3q}{2q}\).
2. To solve for \(q\), eliminate the denominators by cross-multiplying: \(2000 \times 2q = (7000 - 3q)(2000 - q)\).
3. Expand both sides of the equation: On the left, multiply \(2000 \times 2q\); on the right, use the distributive property to expand \((7000 - 3q)(2000 - q)\).
4. After expanding, combine like terms and rearrange the equation to form a quadratic equation in standard form: \(ax^2 + bx + c = 0\).
5. Use the quadratic formula \(q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve for \(q\). Once you find the valid value(s) of \(q\), substitute back into either the supply or demand equation to find the equilibrium price \(p\).

Key Concepts

Equilibrium Price and Quantity

The equilibrium price is the price at which the quantity supplied equals the quantity demanded. This occurs where the supply and demand equations intersect, meaning their price values are equal for the same quantity. Finding this price involves solving the system of equations for p and q.

Solving Rational Equations

Both supply and demand equations are rational expressions involving variables in denominators. To solve for equilibrium, you must set the two expressions equal and manipulate the equation carefully, often by cross-multiplying or clearing denominators, to find the variable values without losing valid solutions.

Interpreting Supply and Demand Functions

Supply and demand functions relate price (p) to quantity (q). Understanding how to interpret these functions helps in setting up the problem correctly. For example, recognizing that p depends on q and that the domain restrictions (like denominators not being zero) affect the solution is essential.