find a linear equation given varibale in a word problem

Question

when a wholesaler sold a certain product at $25 per unit, sales were 800 units per week.  After a price increase of $5 the average number of units sold dropped to 775 units per week.  Assume the price p is a linear function of the demand x.  Write the equation of the price p in terms of the demand x

Answer 1

It seem from the problem statement that the dependent variable (demand x) should be a function of the independent variable (price p).  However, the inverse function of a line is usually also a line, and the problem did say, "Assume the price p is a linear function of the demand x," so let's do that.

 

We have two points:     P1: (800,25)    and P2: (775,30)

The point-slope form of an equation is:    (y-y1) = [(y2-y1)/(x2-x1)] (x - x1)

 

      (p-25) = ( (30-25)/(775-800) )(x - 800)      (plug in points)

      (p-25) = (-1/5)(x-800)

       p = (-1/5)x +185

  

Checking (very important):

   Is  25 = (-1/5)(800) + 185   ?

        25 = -160 + 185  ?

        25 = 25  ?   yes.

   Is  30 = (-1/5)(775) + 185    ?

        30 = -155 + 185   ?

        30 = 30   ?  yes.

Answer 2

Let x = number of units sold if the unit price is p.

 

We are assuming that p is a linear function of x.

 

So, (800, 25) and (775, 30) are two points on a line.

 

slope = (30-25)/(775-800) = 5/(-25) = -1/5

 

Using the point-slope form of the equation of a line, we get                   

  p - 25 = (-1/5)(x - 800)

 

  p - 25 = (-1/5)x + 160

 

          p = (-1/5)x + 185