The given problem involves two equations representing demand and supply functions:

1. $2Q + 4P = 100$ (Demand Function)
2. $3Q - 6P = -60$ (Supply Function)

Instructions:

1. Graph both demand and supply functions along with the equilibrium point: To do this, we need to:
   • Solve both equations for $P$ (price) as a function of $Q$ (quantity).
   • Find the equilibrium point by solving for $Q$ and $P$ when demand equals supply.
   • Draw the graph of both functions and mark the equilibrium point.

Let's first rearrange both equations to express $P$ in terms of $Q$.

Step 1: Demand function

Given: $2Q + 4P = 100$ Solve for $P$: $4P = 100 - 2Q$ $P = \frac{100 - 2Q}{4}$ $P = 25 - 0.5Q$

This is the demand function: $P = 25 - 0.5Q$.

Step 2: Supply function

Given: $3Q - 6P = -60$ Solve for $P$: $-6P = -60 - 3Q$ $P = \frac{60 + 3Q}{6}$ $P = 10 + 0.5Q$

This is the supply function: $P = 10 + 0.5Q$.

Step 3: Find the equilibrium point

At equilibrium, the quantity demanded equals the quantity supplied, i.e., the two price equations are equal: $25 - 0.5Q = 10 + 0.5Q$

Solving for $Q$: $25 - 10 = 0.5Q + 0.5Q$ $15 = Q$

Now, substitute $Q = 15$ into either equation to find $P$. Using the demand function: $P = 25 - 0.5(15) = 25 - 7.5 = 17.5$

Thus, the equilibrium point is $(Q, P) = (15, 17.5)$.

Step 4: Graph the functions

• The demand function is a downward-sloping line: $P = 25 - 0.5Q$.
• The supply function is an upward-sloping line: $P = 10 + 0.5Q$.
• The equilibrium point is $(15, 17.5)$.

I can create this graph for you. Would you like me to generate it?

Would you like more details, or do you have any questions about the steps? Here are some related questions for practice:

1. How would the graph change if the demand function's slope increased?
2. What happens to the equilibrium price and quantity if the supply function shifts downward?
3. How can we model a tax imposed on the supplier in this model?
4. What is the elasticity of demand in this scenario?
5. How does the equilibrium adjust if both demand and supply functions change?

Tip: When graphing linear functions like these, always check how the slope and intercept define the function's behavior over the range of $Q$.