# Supply and demand

The goal is to find supply and demand equations using some given information and then use the equations to find equilibrium point.

After doing some market research, a manufacturer notices the following pattern for selling an item. The graph for the following situation is shown below.

Information about the supply based on the price

Price                             Supply

1 dollar                         600 units
4  dollars                      4200 units
6  dollars                      6600 units
9 dollars                       10200 units

Information about the demand based on the price

Price                                Demand

1 dollar                            3600 units
4 dollars                          900 units
6 dollars                          600 units
9 dollars                          400 units Notice that for the supply, as the price goes up, the number of items goes up too. This is so because it will cost the manufacturer more money to produce more items.

However, for the demand, as the price goes up, the number of items goes down. People have a tendency to buy less when the price goes up.

## Supply and demand equations

Use the information above to find the supply and demand equations.

How to find the demand equation

Usually, the demand equation is modeled with an inverse variation.

The inverse variation equation is y =
k / x

Pick (9, 400) to find k although you can pick something else such as (1, 3600)

400 =
k / 9

Since 400 =
3600 / 9
, k = 3600

y =
3600 / x

In terms of demand (d) and price (p), we get:

Demand equation

d =
3600 / p

How to find the supply equation

Usually, the supply equation is modeled with a linear equation.

The linear equation is y = mx + b

Use (4, 4200) and (9, 10200) to find m

m =
10200 - 4200 / 9 - 4

m =
6000 / 5
= 1200

y = 1200x + b

Use (4, 4200) to find b

4200 = 1200 × 4 + b

4200 = 4800 + b

4200 - 4800 = b

-600 = b

y = 1200x + -600

In terms of p and supply ( s ), we get

Supply equation:

s = 1200p + -600

How to find the equilibrium point

The equilibrium point is the price at which the supply is equal to the demand

3600 / p
= 1200p + -600

Multiply both sides by p

p ×
3600 / p
= p × (1200p + -600)

3600 = 1200p2 + -600p

Divide both sides by 600

6 = 2p2 + -p

2p2 + -p - 6 = 0

(p - 2) × (2p + 3) = 0

p = 2 and p = -3/2

p = 2 since the price must be positive

The equilibrium point is 2 dollars.

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