At a price of 160 soms, 2 tons of cookies are offered on the market, at a price of 135 soms there are 1.6 tons. Draw a line through these points. Write the cookies supply linear function as the price function of quantity. At a price of 175 soms, 1.42 tons of cookies are offered on the market, at a price of 153 soms there are 1.97 tons. Draw a line through these points. Write the demand linear function for cookies as the price function of quantity. Determine the equilibrium price and equilibrium quantity for this market

To solve this problem, we need to find the supply and demand linear functions based on the provided data points, and then find the equilibrium price and quantity where the supply and demand intersect.

### Step 1: Determine the Supply Function

We are given two points for the supply:

1. At a price of 160 soms, 2 tons of cookies are supplied.
2. At a price of 135 soms, 1.6 tons of cookies are supplied.

These points can be represented as \((q_1, p_1) = (2, 160)\) and \((q_2, p_2) = (1.6, 135)\), where \(q\) is the quantity in tons and \(p\) is the price in soms.

We assume that the supply function is linear, of the form:

\[
p = m_s \cdot q + b_s
\]

To find the slope \(m_s\), we use the formula:

\[
m_s = \frac{p_2 - p_1}{q_2 - q_1}
\]

Substituting the given points:

\[
m_s = \frac{135 - 160}{1.6 - 2} = \frac{-25}{-0.4} = 62.5
\]

Now, use the point-slope form to find \(b_s\). We can use point \((2, 160)\):

\[
160 = 62.5 \cdot 2 + b_s
\]

\[
160 = 125 + b_s
\]

\[
b_s = 35
\]

So, the supply function is:

\[
p = 62.5q + 35
\]

### Step 2: Determine the Demand Function

We are given two points for the demand:

1. At a price of 175 soms, 1.42 tons of cookies are demanded.
2. At a price of 153 soms, 1.97 tons of cookies are demanded.

These points can be represented as \((q_3, p_3) = (1.42, 175)\) and \((q_4, p_4) = (1.97, 153)\).

We assume that the demand function is also linear, of the form:

\[
p = m_d \cdot q + b_d
\]

To find the slope \(m_d\), we use the formula:

\[
m_d = \frac{p_4 - p_3}{q_4 - q_3}
\]

Substituting the given points:

\[
m_d = \frac{153 - 175}{1.97 - 1.42} = \frac{-22}{0.55} = -40
\]

Now, use the point-slope form to find \(b_d\). We can use point \((1.42, 175)\):

\[
175 = -40 \cdot 1.42 + b_d
\]

\[
175 = -56.8 + b_d
\]

\[
b_d = 231.8
\]

So, the demand function is:

\[
p = -40q + 231.8
\]

### Step 3: Determine the Equilibrium Price and Quantity

The equilibrium occurs where the supply equals the demand, so we set the supply and demand functions equal to each other:

\[
62.5q + 35 = -40q + 231.8
\]

Combine like terms:

\[
62.5q + 40q = 231.8 - 35
\]

\[
102.5q = 196.8
\]

Solve for \(q\):

\[
q = \frac{196.8}{102.5} \approx 1.92
\]

Now, substitute \(q = 1.92\) back into either the supply or demand function to find the equilibrium price \(p\). Using the supply function:

\[
p = 62.5 \cdot 1.92 + 35
\]

\[
p = 120 + 35 = 155
\]

### Conclusion

- **Equilibrium Quantity**: \(q = 1.92\) tons
- **Equilibrium Price**: \(p = 155\) soms

These values represent the point where the quantity of cookies supplied equals the quantity demanded in the market.

Would you like more details or have any questions? Here are five related questions that expand on this topic:

1. How do changes in the slope of the supply and demand functions affect the equilibrium?
2. What happens to the equilibrium if there is a sudden increase in supply?
3. How do we interpret the slopes of the supply and demand functions in economic terms?
4. How can we use this linear model to predict changes in supply and demand?
5. How would a tax on cookies affect the supply and demand functions?

**Tip**: Understanding how to find the equilibrium helps in analyzing market conditions and predicting the impact of various economic policies.

At a price of 160 soms, 2 tons of cookies are offered on the market, at a price of 135 soms there are 1.6 tons. Draw a line through these points. Write the cookies supply linear function as the price function of quantity. At a price of 175 soms, 1.42 tons of cookies are offered on the market, at a price of 153 soms there are 1.97 tons. Draw a line through these points. Write the demand linear function for cookies as the price function of quantity. Determine the equilibrium price and equilibrium quantity for this market.

Learn how to calculate the supply and demand linear functions for cookies and find the equilibrium price and quantity. This problem involves key concepts like linear functions, slope calculations, and determining the intersection of supply and demand. Ideal for Grades 9-12.

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