Question

A company makes t-shirts and their research shows that that price and demand are related linearly: p = m x + b . They know that in order to sell 5 shirts they need to set the price at $50, and in order to sell 10 shirts they need to set the price at $40. Find the linear equation relating price to demand.

Answer 1

The linear equation relating price to demand is Q=-0.5*P + 30

Explanation:

The demand for a good is the quantity of that good that the demanders are willing to purchase at a given price.

Then, the Demand Curve relates the quantity that a consumer would be willing to buy as a function of price.

You seek to determine the linear equation that relates the price to the demand Q = m * P + b where Q is the quantity demanded at a price P, and compared with the equation of the straight line y = mx + b, m is the slope and b is the ordinate to the origin.

On a line of the form y = m * x + b, the value of m, having two points, is calculated by:

`m=(y2-y1)/(x2-x1)`

And the ordinate to the origin b is calculated, taking the value of m, replacing any of the two points (replacing any of the two because it must give the same value of b) in the expression y = m * x + b and solving its value.

In the case of Q=m*P+b, the value of m, having two points, is calculated by:

`m=(Q2-Q1)/(P2-P1)`

Being:

• Q1= 5
• P1= $50
• Q2= 10
• P2= $40

and replacing:

`m=(10-5)/(40-50)`

you get:

m= -0.5

So, being Q=-0.5*P+b, b is calculated by:

• Replacing the point Q1= 5 and P1= 50

5= -0.5*50 + b → 5= -25 + b → 5+25= b → b= 30

• Replacing the point Q1= 10 and P1= 40

10= -0.5*40 + b → 10= -20 + b → 10+20= b → b= 30

Then, the linear equation relating price to demand is Q=-0.5*P + 30