Let's solve the two problems given in the image:

Problem 1:

The supply function is $P = 7 + Q$ and the demand function is $P = 16 - 2Q$.

(a) Find the equilibrium price and quantity:

At equilibrium, the quantity supplied equals the quantity demanded. Thus, we set the two equations equal to each other:

$7 + Q = 16 - 2Q$

Solving for $Q$:

$7 + Q = 16 - 2Q$ $Q + 2Q = 16 - 7$ $3Q = 9$ $Q = 3$

Now, substitute $Q = 3$ into either the supply or demand function to find the equilibrium price $P$.

Using the supply function $P = 7 + Q$:

$P = 7 + 3 = 10$

Thus, the equilibrium price is $P = 10$, and the equilibrium quantity is $Q = 3$.

(b) Graph the curves:

1. Supply curve $P = 7 + Q$ is a straight line with a slope of 1 and a y-intercept of 7.
2. Demand curve $P = 16 - 2Q$ is a straight line with a slope of -2 and a y-intercept of 16.

The intersection point of these two lines occurs at $P = 10$ and $Q = 3$, which represents the equilibrium point.

Problem 2:

The demand function is $P = 24 - 3Q^2$ and the supply function is $P = Q^2 + 2Q + 4$.

Find the equilibrium price and quantity:

Set the two equations equal to find the equilibrium:

$24 - 3Q^2 = Q^2 + 2Q + 4$

Rearranging terms:

$24 - 3Q^2 = Q^2 + 2Q + 4$ $0 = 4Q^2 + 2Q + (4 - 24)$ $0 = 4Q^2 + 2Q - 20$ $0 = 2Q^2 + Q - 10$

Now, solve this quadratic equation using the quadratic formula:

$Q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a = 2$, $b = 1$, and $c = -10$. Substituting these values:

$Q = \frac{-1 \pm \sqrt{1^2 - 4(2)(-10)}}{2(2)}$ $Q = \frac{-1 \pm \sqrt{1 + 80}}{4}$ $Q = \frac{-1 \pm \sqrt{81}}{4}$ $Q = \frac{-1 \pm 9}{4}$

Thus, the two solutions are:

$Q = \frac{-1 + 9}{4} = \frac{8}{4} = 2$ $Q = \frac{-1 - 9}{4} = \frac{-10}{4} = -2.5$

Since quantity $Q$ cannot be negative, we take $Q = 2$.

Now, substitute $Q = 2$ into either the supply or demand function to find the equilibrium price $P$.

Using the demand function $P = 24 - 3Q^2$:

$P = 24 - 3(2)^2 = 24 - 3(4) = 24 - 12 = 12$

Thus, the equilibrium price is $P = 12$, and the equilibrium quantity is $Q = 2$.

Summary of Solutions:

1. For Problem 1:

   • Equilibrium price: $P = 10$
   • Equilibrium quantity: $Q = 3$

2. For Problem 2:

   • Equilibrium price: $P = 12$
   • Equilibrium quantity: $Q = 2$

Would you like further details or have any questions about these solutions?

Here are some related questions:

1. How do changes in the demand curve affect the equilibrium?
2. How does the supply curve's slope influence market behavior?
3. Can equilibrium prices change if there are shifts in external factors like taxes or subsidies?
4. What happens to equilibrium when both supply and demand curves shift?
5. How can you determine the consumer and producer surplus at equilibrium?

Tip: To solve equilibrium problems efficiently, always set the supply and demand functions equal and solve for the quantity first!