SOLUTION: Suppose you have a lemonade stand, and when you charge $2 per cup of lemonade you sell 120 cups. But when you raise your price to $3 you only sell 60 cups. Write an equation for

Question 107873: Suppose you have a lemonade stand, and when you charge $2 per cup of lemonade you sell 120 cups. But when you raise your price to $3 you only sell 60 cups. Write an equation for the number of cups you sell as a function of the price you charge. Denote "C" for number of cups, and "P" for the price you charge. Assume the function is linear.

This is what I have so far, please help!
Linear function: C(P)=Pm+b
Price- $2, cups sold- 120
Price- $3, cups sold- 60
Take 120 from 60 we get a= -60

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Let x=price, y=# of cups sold

So when x=2, then y=120 (ie you sell 120 cups at $2). So that means you have the point (2,120). Now let (,) be that first point (ie and )

Also when x=3, then y=60 (ie you sell 60 cups at $3). So that means you have the point (3,60). Now let (,) be the second point (ie and )

So we have two points (2,120) and (3,60)

Now let's find the equation of the line through those points

First lets find the slope through the points (,) and (,)

Start with the slope formula (note: (,) is the first point (,) and (,) is the second point (,))

Plug in ,,, (these are the coordinates of given points)

Subtract the terms in the numerator to get . Subtract the terms in the denominator to get

So the slope is

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Now let's use the point-slope formula to find the equation of the line:

------Point-Slope Formula------
where is the slope, and (,) is one of the given points

So lets use the Point-Slope Formula to find the equation of the line

Plug in , , and (these values are given)

Distribute

Multiply and to get

Add to both sides to isolate y

Combine like terms and to get
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Answer:

So the equation of the line which goes through the points (,) and (,) is:

The equation is now in form (which is slope-intercept form) where the slope is and the y-intercept is

Notice if we graph the equation and plot the points (,) and (,), we get this: (note: if you need help with graphing, check out this solver)

Graph of through the points (,) and (,)

Notice how the two points lie on the line. This graphically verifies our answer.

Since the equation is the function is where is the cost (relative to a given price) and p is the price
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