Answer:
p = - 2x + 54
Step-by-step explanation:
The price-demand linear equation is
p = mx + b
which is the equation of a straight line in slope-intercept form
The parameters here are m = slope and b = y-intercept in line equation terms
x is the number of shirts sold and p is the price
Using values given, we can generate two equations which can solve for the parameters of the linear equation
To sell 10 shirts, the price has to be $34
==> 34 = m x 10 + b = 10m + b
Switch sides to get 10m + b = 34 (1)
To sell 20 shirts, the price has to be $14
==> 14 = 20m + b
Switch sides to get 20m + b = 14 (2)
Subtract (2) from (1) to eliminate the term b
(1) - (2)
==> 10m+b - (20m+b) = 34 - 14
==> 10m + b - 20m - b = 20
==> 10m - 20m = 20
==> - 10m = 20
==> m = 20/(- 10) = - 2.4
==> m = - 2.0
This term is also referred to as the elasticity of demand ie how much demand changes with respect to price change. It is negative here because as you increase the price, the demand decreases
Now that we have value for m all we have to do is plug it into one of the equations (1) or (2) and solve for b
Using equation (2),
20(- 2) + b = 14
==> - 40 + b = 14
==> 0 + b = 14 + 54(Add 48 on both sides)
==> b = 54
So the price-demand equation is
[tex]\boxed{\sf{p=-2x + 54}}[/tex]
