linear equation

Question

1. It is estimated that if a price of $15 per unit is charged for a product, five units per period will be sold. If the price is lowered to $13, sales will be six units per period.

The supply curve for the product is given by P = 3Q + 5

                                                                                   = 3+0

                                                                                   = 3#

a) Assuming that the demand relationship is linear, find the demand equation.

b) Find the equilibrium price and quantity (i.e. the price where quantity demanded = quantity supplied).

Answer 1

Let’s solve this step by step.

a) Finding the Demand Equation

1. Identify Points:
2. Point 1: Price = 15, Quantity = 5 (whenpriceis 15, sell 5 units)
3. Point 2: Price = 13, Quantity = 6 (whenpriceis 13, sell 6 units)
4. Calculate the Slope (m):

m = (P₂ – P₁)

(Q₂ – Q₁)

m = (13 – 15) = –2⁄1 = – 2

( 6 – 5 )

Use Point-Slope Form to Write Equation:

Formula P – P₁ = m ( Q – Q₁)

Choose Point 1 (5, 15):

P −15 = −2 (Q−5)

Simplify:

P −15 = − 2Q + 10

P = −2Q + 25

So, the demand equation is:

P = −2Q + 25

b) Finding Equilibrium Price and Quantity

Set Demand Equal to Supply:

Demand: P = −2Q +25

Supply: P = 3Q + 5

Set Equations Equal:

−2Q + 25 = 3Q + 5

Solve for Q:

Add 2Q to both sides:

25 = 5Q +5

Subtract 5 from both sides:

20 = 5Q

Divide by 5:

Q = 4

Find Equilibrium Price:

Substitute Q = 4 into either equation (use supply):

P = 3 (4) + 5 = 12 + 5 = 17

So, the equilibrium price is $17 and the equilibrium quantity is 4 units.