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 demand.

Verified step by step guidance

1. Identify the equilibrium condition: At equilibrium, the supply price equals the demand price. So, set the supply equation equal to the demand equation: \(\frac{2000}{2000 - q} = \frac{7000 - 3q}{2q}\).
2. Clear the denominators by cross-multiplying to eliminate the fractions: \(2000 \times 2q = (7000 - 3q)(2000 - q)\).
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. Simplify the resulting equation by combining like terms and rearranging all terms to one side to form a polynomial equation in terms of \(q\).
5. Solve the polynomial equation for \(q\) to find the equilibrium demand. Remember to check for any extraneous solutions that might not make sense in the context of the problem.

Key Concepts

Equilibrium in Supply and Demand

Equilibrium occurs when the quantity supplied equals the quantity demanded, meaning the supply and demand equations have the same price (p) for a certain quantity (q). Finding equilibrium involves setting the supply price equal to the demand price and solving for q.

Solving Rational Equations

Both supply and demand equations are rational expressions involving q in denominators. To solve for q, you must manipulate these fractions carefully, often by cross-multiplying or finding a common denominator, to form a solvable algebraic equation.

Interpreting Variables in Context

Understanding that p represents price and q represents quantity is essential. This context helps interpret the solution meaningfully, ensuring that the equilibrium quantity found is realistic and applicable to the commodity market described.