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To solve this problem step by step, we need to find the equilibrium price and quantity for the given supply and demand functions. The demand function is \( Q_d = 230 - 4P \) and the supply function is \( Q_s = -50 + 3P \). Let’s break down the solution:

### Step 1: Calculate the Equilibrium Price and Quantity
The equilibrium occurs when the quantity demanded (\( Q_d \)) equals the quantity supplied (\( Q_s \)). So, we set the two equations equal to each other:

\[
230 - 4P = -50 + 3P
\]

Now, solve for \( P \) (the equilibrium price):

1. Move the terms involving \( P \) to one side:

\[
230 + 50 = 3P + 4P
\]

\[
280 = 7P
\]

2. Solve for \( P \):

\[
P = \frac{280}{7} = 40
\]

So, the equilibrium price \( P_e \) is 40.

### Step 2: Calculate the Equilibrium Quantity
Now, substitute \( P = 40 \) back into either the demand or supply function to find the equilibrium quantity \( Q_e \). Let’s use the demand function:

\[
Q_d = 230 - 4(40) = 230 - 160 = 70
\]

So, the equilibrium quantity \( Q_e \) is 70.

### Step 3: Graph the Demand and Supply Functions
The demand function is \( Q_d = 230 - 4P \), which is a linear function with a negative slope. The supply function is \( Q_s = -50 + 3P \), which is a linear function with a positive slope.

- **Demand Function Graph**: It starts from \( Q_d = 230 \) when \( P = 0 \), and the slope is \( -4 \).
- **Supply Function Graph**: It starts from \( Q_s = -50 \) when \( P = 0 \), and the slope is \( 3 \).

These two lines intersect at the equilibrium point \( (P_e = 40, Q_e = 70) \).

### Step 4: Explain the Gradient (Slope)
The gradient (or slope) in these linear functions represents the rate of change of quantity with respect to price:

- **Demand Function Slope**: The slope is \( -4 \), meaning that for every 1 unit increase in price, the quantity demanded decreases by 4 units. This reflects the law of demand.
- **Supply Function Slope**: The slope is \( 3 \), meaning that for every 1 unit increase in price, the quantity supplied increases by 3 units. This reflects the law of supply.

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Would you like a graph illustrating these functions? Let me know if you need any further clarifications or a more detailed explanation.

### Follow-up Questions:
1. How does the equilibrium change if the supply or demand function changes?
2. Can you explain why the demand function has a negative slope?
3. What happens to the equilibrium if there is a shift in the demand curve?
4. How would you interpret a shift in the supply curve for this model?
5. What real-world factors could affect the supply or demand in this model?

#### Tip:
To draw the graph of supply and demand more easily, first plot the points where the lines intersect the axes, then connect them to the equilibrium point.

Diketahui fungsi permintaan suatu barang adalah Qd = 230 - 4P dan fungsi penawarannya adalah Qs = -50 + 3P. Hitunglah harga dan jumlah keseimbangan.

This problem involves calculating the equilibrium price and quantity from given linear demand and supply functions, using algebraic methods to solve the system of equations. Suitable for Grades 10-12.

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