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Let's begin by analyzing the demand and supply functions provided in the problem:

- The demand function is given as:
  \[
  Q_d = 230 - 4P
  \]
  where \(Q_d\) is the quantity demanded and \(P\) is the price.

- The supply function is given as:
  \[
  Q_s = -50 + 3P
  \]
  where \(Q_s\) is the quantity supplied.

### 1. **Finding the equilibrium price and quantity:**
At equilibrium, the quantity demanded equals the quantity supplied, i.e.:
\[
Q_d = Q_s
\]
Substitute the given functions into this equation:
\[
230 - 4P = -50 + 3P
\]
Solve for \(P\) (the equilibrium price):

\[
230 + 50 = 3P + 4P
\]
\[
280 = 7P
\]
\[
P = \frac{280}{7} = 40
\]
So, the equilibrium price is **40**.

Next, substitute \(P = 40\) into either the demand or supply function to find the equilibrium quantity:

Using the demand function:
\[
Q_d = 230 - 4(40) = 230 - 160 = 70
\]
Thus, the equilibrium quantity is **70**.

### 2. **Graphing the demand and supply curves:**
The demand curve is derived from the equation \(Q_d = 230 - 4P\), and the supply curve from \(Q_s = -50 + 3P\).

To sketch the curves, consider the intercepts:
- For the demand curve, set \(P = 0\) to find the \(Q_d\)-intercept:
  \[
  Q_d = 230 - 4(0) = 230
  \]
  Set \(Q_d = 0\) to find the price intercept:
  \[
  0 = 230 - 4P \implies P = \frac{230}{4} = 57.5
  \]
- For the supply curve, set \(P = 0\) to find the \(Q_s\)-intercept:
  \[
  Q_s = -50 + 3(0) = -50
  \]
  Set \(Q_s = 0\) to find the price intercept:
  \[
  0 = -50 + 3P \implies P = \frac{50}{3} \approx 16.67
  \]

With these points, you can now sketch the curves on a graph, showing the demand curve with a negative slope and the supply curve with a positive slope.

### 3. **Explaining the gradients (slopes) of the functions:**
- The gradient of the demand function is \(-4\). This means that for every 1-unit increase in price, the quantity demanded decreases by 4 units. A negative slope indicates an inverse relationship between price and quantity demanded.
  
- The gradient of the supply function is \(+3\). This means that for every 1-unit increase in price, the quantity supplied increases by 3 units. A positive slope indicates a direct relationship between price and quantity supplied.

Would you like a detailed graphical representation, or is there a specific part of the solution you'd like more detail on?

### Additional questions:
1. What happens to equilibrium if the demand curve shifts?
2. How does a change in the slope of the supply curve affect the equilibrium?
3. What real-world factors might cause the supply or demand functions to shift?
4. How would taxes affect the equilibrium price and quantity in this scenario?
5. Can you calculate the consumer surplus and producer surplus at equilibrium?

**Tip:** The gradients in demand and supply functions are crucial for determining the sensitivity of quantity to changes in price (price elasticity).

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

Learn how to find the equilibrium price and quantity using linear demand and supply functions. The problem involves solving for price and quantity where the demand Qd = 230 - 4P and supply Qs = -50 + 3P are equal. Suitable for students in Grades 10-12.

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