The Unique Gifts catalog lists a "super loud and vibrating alarm clock." Their records indicate the following information on the relation of monthly supply and demand quantities to the price of the clock.

Demand    Supply    Price

168

133

$31

138

173

$55

Use this information to find the following.

(a) points on the demand linear equation

(*x*, *p*) =

(smaller *x*-value)

(*x*, *p*) =

(larger *x*-value)

points on the supply linear equation

(*x*, *p*) =

(smaller *x*-value)

(*x*, *p*) =

(larger *x*-value)

(b) the demand equation *p*
*p* =

(c) the supply equation *p*
*p* =

(d) the equilibrium quantity and price

Equilibrium occurs when the price of the clock is $  and the quantity is

To solve this problem, we will first address the information provided and calculate the demand and supply linear equations, followed by determining the equilibrium price and quantity.

### (a) Points on the Demand and Supply Linear Equations

#### Demand Data:
We are given two sets of values for demand and price:

- When demand is 168, the price is $31.
- When demand is 138, the price is $55.

Thus, we have two points on the demand curve:  
$(168, 31)$ and $(138, 55)$.

#### Supply Data:
We are also given two sets of values for supply and price:

- When supply is 133, the price is $31.
- When supply is 173, the price is $55.

Thus, we have two points on the supply curve:  
$(133, 31)$ and $(173, 55)$.

### (b) Demand Equation

To find the demand equation, we first calculate the slope of the demand line using the two points $(168, 31)$ and $(138, 55)$:

\[
m_{\text{demand}} = \frac{55 - 31}{138 - 168} = \frac{24}{-30} = -0.8
\]

Now we use the point-slope form of a linear equation $p - p_1 = m(x - x_1)$ to find the demand equation:

Using point $(168, 31)$ with $m = -0.8$:

\[
p - 31 = -0.8(x - 168)
\]

Simplify this equation:

\[
p - 31 = -0.8x + 134.4
\]

\[
p = -0.8x + 165.4
\]

Thus, the **demand equation** is:

\[
p = -0.8x + 165.4
\]

### (c) Supply Equation

Next, we calculate the slope of the supply line using the two points $(133, 31)$ and $(173, 55)$:

\[
m_{\text{supply}} = \frac{55 - 31}{173 - 133} = \frac{24}{40} = 0.6
\]

Now, using the point-slope form with point $(133, 31)$ and $m = 0.6$:

\[
p - 31 = 0.6(x - 133)
\]

Simplify this equation:

\[
p - 31 = 0.6x - 79.8
\]

\[
p = 0.6x - 48.8
\]

Thus, the **supply equation** is:

\[
p = 0.6x - 48.8
\]

### (d) Equilibrium Quantity and Price

Equilibrium occurs when the demand price equals the supply price, i.e., the two equations are equal:

\[
-0.8x + 165.4 = 0.6x - 48.8
\]

Solve for $x$ (quantity):

\[
165.4 + 48.8 = 0.6x + 0.8x
\]

\[
214.2 = 1.4x
\]

\[
x = \frac{214.2}{1.4} = 153
\]

Now, substitute $x = 153$ into either equation to find the equilibrium price. Using the demand equation:

\[
p = -0.8(153) + 165.4 = -122.4 + 165.4 = 43
\]

Thus, the **equilibrium quantity** is 153, and the **equilibrium price** is $43.

### Summary:

1. **Points on the demand linear equation:**
   - $(168, 31)$ and $(138, 55)$
   
2. **Points on the supply linear equation:**
   - $(133, 31)$ and $(173, 55)$

3. **Demand equation:**
   \[
   p = -0.8x + 165.4
   \]

4. **Supply equation:**
   \[
   p = 0.6x - 48.8
   \]

5. **Equilibrium quantity and price:**
   - Equilibrium quantity: 153 units
   - Equilibrium price: $43

---

Would you like more details or have any further questions? 

Here are some related questions:
1. How do linear supply and demand curves reflect market behavior?
2. Can we calculate a price elasticity of demand from this data?
3. What happens if a price ceiling is set below the equilibrium price?
4. How do changes in production cost affect the supply equation?
5. How would a tax on the product affect both supply and demand?

**Tip:** When finding equilibrium, you solve by setting the demand and supply functions equal, representing the point where the market clears.

The Unique Gifts catalog lists a 'super loud and vibrating alarm clock.' Their records indicate the following information on the relation of monthly supply and demand quantities to the price of the clock.

Demand    Supply    Price
168       133       $31
138       173       $55

Use this information to find the following.

(a) points on the demand linear equation

(*x*, *p*) = (smaller *x*-value)
(*x*, *p*) = (larger *x*-value)

points on the supply linear equation

(*x*, *p*) = (smaller *x*-value)
(*x*, *p*) = (larger *x*-value)

(b) the demand equation *p*
*p* =

(c) the supply equation *p*
*p* =

(d) the equilibrium quantity and price

Equilibrium occurs when the price of the clock is $  and the quantity is

Learn how to find the linear equations for supply and demand based on price points for a 'super loud and vibrating alarm clock.' This example covers key concepts like slope-intercept form, point-slope equations, and calculating the equilibrium price and quantity. Suitable for students in Grades 9-11.

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