Market data indicates customers will buy 700 sprockets if they are prices at $13 each. If the price rises to $15 they will only buy 500. If the price is $12 a sprocket, the producers will make 400 sprockets. If the price rises to $13 a sprocket, they will produce 600 sprockets. (a) Find a linear model for the demand price. (b) Find a linear model for the supply price. (c) Use Desmos to find the equilibrium point. What number of units is supplied and demanded? What is the price?

Demand = -100P + 2,000

Supply = 200P - 2200

Equilibrium: 600 units and $14 price

Explanation:

We solve for the linear formula of each:

Demand = aP + b

       Demand

  Quantity  Price

     700        13

     500        15

(700 - 500) / (13 - 15) = 200 / -2 = -100

a = -100

(-100) x 13  + b = 700

(-100) x 15  + b = 500

b = 700 + 1,300 =  2,000

b = 500 + 1,500 = 2,000

Demand = -100P + 2,000

       Supply

 Quantity   price

   400        12

   600        13

Supply = aP + b

(600 - 400) / ( 13 - 12) = 200

a = 200

400 = 200 x $13 + b

b = 400 + 2600 = -2200

Supply = 200P - 2200

Equilibrium

200P - 2200 = -100P + 2000

300P = 4200

P = 4200 / 300 = 14

Q = -100(14) + 2000 = 600

200 x (14) - 2200 = 600

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