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Let's solve the given system of equations step by step.

### Given equations:
- \( Q_{d1} = 18 - 3P_1 + P_2 \) (Demand for commodity 1)
- \( Q_{s1} = -2 + 4P_1 \) (Supply for commodity 1)
- \( Q_{d2} = 12 + P_1 - 2P_2 \) (Demand for commodity 2)
- \( Q_{s2} = -2 + 3P_2 \) (Supply for commodity 2)

At equilibrium, the quantity demanded equals the quantity supplied for both commodities:
- \( Q_{d1} = Q_{s1} \)
- \( Q_{d2} = Q_{s2} \)

### Step 1: Solve for \( P_1 \) and \( P_2 \)

#### First equation (equating \( Q_{d1} \) and \( Q_{s1} \)):
\[
18 - 3P_1 + P_2 = -2 + 4P_1
\]
Simplify:
\[
18 + P_2 = -2 + 7P_1
\]
\[
20 + P_2 = 7P_1 \quad \text{(Equation 1)}
\]

#### Second equation (equating \( Q_{d2} \) and \( Q_{s2} \)):
\[
12 + P_1 - 2P_2 = -2 + 3P_2
\]
Simplify:
\[
14 + P_1 = 5P_2 \quad \text{(Equation 2)}
\]

### Step 2: Solve the system of equations

From Equation (2):
\[
P_1 = 5P_2 - 14
\]
Substitute this into Equation (1):
\[
20 + P_2 = 7(5P_2 - 14)
\]
\[
20 + P_2 = 35P_2 - 98
\]
Simplify:
\[
118 = 34P_2
\]
\[
P_2 = \frac{118}{34} = \frac{59}{17}
\]

Now, substitute \( P_2 = \frac{59}{17} \) into the expression for \( P_1 \):
\[
P_1 = 5\left(\frac{59}{17}\right) - 14 = \frac{295}{17} - 14 = \frac{295}{17} - \frac{238}{17} = \frac{57}{17}
\]

### Step 3: Find equilibrium quantities \( Q_1 \) and \( Q_2 \)

#### For \( Q_1 \):
Substitute \( P_1 = \frac{57}{17} \) and \( P_2 = \frac{59}{17} \) into the equation for \( Q_{d1} \):
\[
Q_{d1} = 18 - 3P_1 + P_2 = 18 - 3\left(\frac{57}{17}\right) + \frac{59}{17}
\]
\[
Q_{d1} = 18 - \frac{171}{17} + \frac{59}{17} = 18 - \frac{112}{17} = 18 - 6.588 = 11.412
\]
Thus, \( Q_1 = 11.412 \).

#### For \( Q_2 \):
Substitute \( P_1 = \frac{57}{17} \) and \( P_2 = \frac{59}{17} \) into the equation for \( Q_{d2} \):
\[
Q_{d2} = 12 + P_1 - 2P_2 = 12 + \frac{57}{17} - 2\left(\frac{59}{17}\right)
\]
\[
Q_{d2} = 12 + \frac{57}{17} - \frac{118}{17} = 12 - \frac{61}{17} = 12 - 3.588 = 8.412
\]
Thus, \( Q_2 = 8.412 \).

### Summary of equilibrium values:
- \( P_1 = \frac{57}{17} \)
- \( P_2 = \frac{59}{17} \)
- \( Q_1 = 11.412 \)
- \( Q_2 = 8.412 \)

---

### (ii) Relationship between the two commodities:
The prices of the two commodities are interdependent, as shown by how the price of commodity 1 (\( P_1 \)) affects the demand for commodity 2 (\( Q_{d2} \)), and vice versa for commodity 1. Specifically, the price of one commodity appears in the demand equation of the other, indicating that the two commodities are either complements or substitutes.

Would you like more details or have any questions?

### Related questions:
1. How do we classify complementary and substitute goods in economics?
2. Can we interpret the relationship between two goods from their demand equations alone?
3. How would the equilibrium change if one of the commodities becomes more expensive?
4. What if the supply functions were more elastic—how would that affect equilibrium?
5. How would a shift in consumer preference impact the equilibrium in this model?

**Tip**: In equilibrium analysis, always check how the variables (price, quantity) of one commodity affect the other to understand market interdependence.

The demand and supply functions of a two-commodity market model are as follows:
Qa = 18 - 3P1 + P2 and Qy = -2 + 4P1
Qu = 12 + P1 -2P2 and Qz = -2 + 3P2
(i) Find the equilibrium value of Q1, Q2, P1 and P2.
(ii) How are these two commodities related? Explain.

Analyze the equilibrium values of Q1, Q2, P1, and P2 for a two-commodity market model with given demand and supply functions. Learn how these commodities are interrelated based on their supply and demand equations.

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