SOLUTION: This is a word problem that I am having difficulty with. Suppose a market research company finds that at a price of p= $25, they would sell x= 60 shirts each month. If they low

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: This is a word problem that I am having difficulty with. Suppose a market research company finds that at a price of p= $25, they would sell x= 60 shirts each month. If they low      Log On


   

Question 344768: This is a word problem that I am having difficulty with.
Suppose a market research company finds that at a price of p= $25, they would sell x= 60 shirts each month. If they lower the price to p=$15, then more people would purchase the shirts, and they can expect to sell x=75 shirts in a months time.
I need to find the equation of the line for the demand equation.
Any help would be greatly appreciated!
Thank you,
Shaun
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
At its simplest, this problem boils down to: Given two points, find the equation of the line through those points. Using "p" for "y", the two points are (60, 25) and (75, 15).

To find the equation of a line from two points we usually start by using the slope formula: m+=+%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 to find the slope:
Substituting the coordinates we have into this formula we get:
m+=+%2825-15%29%2F%2860-75%29+=+10%2F%28-15%29+=+%28-2%29%2F3
Now we have the slope of the line and two points on the line. From here there are two methods that are used to get the equation of the line. Not knowing which way you may know, I'll explaing both.

Using the Point-Slope form. The point-slope form of the equation of a line is:
y+-+y%5B1%5D+=+m%28x-x%5B1%5D%29
Into this we can substitute the slope and the coordinates of either point. (It doesn't matter which point you use!):
y+-+25+=+%28%28-2%29%2F3%29%28x+-+60%29
When asked for "the" equation of a line, it is meant the slope-intercept form (y = mx + b). So we will use some basic Algebra to transform the point-slope form into the slope-intercept form. First we'll use the Distributive Property to multiply out the right side:
y+-+25+=+%28%28-2%29%2F3%29x+%2B+40
Now we'll add 25 to each side:
y+=+%28%28-2%29%2F3%29x+%2B+65


Using the slope-intercept form (twice). We can use the slope-intercept form, the slope and one of the points (again, it doesn't matter which one) to find "b" (the y-intercept). We replace the x and y of the slope intercept form with the coordinates of a point and replace the m with the slope. (To show you that it doesn't matter which point, I'm going to use the "other" point this time.)
15+=+%28%28-2%29%2F3%29%2875%29+%2B+b
Now we solve for b. Start by simplifying the right side:
15+=+-50+%2B+b
Add 50 to each side:
65+=+b
Now we have the y-intercept and the other two points and the slope of the line. We can use the slope-intercept form again to find the equation of the line. This time we just replace the slope and the y-intercept:
y+=+%28%28-2%29%2F3%29x+%2B+65

Note how we ended up with the same equation using either method. The only thing left to do is to replace the "y" with a "p":
p+=+%28%28-2%29%2F3%29x+%2B+65