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

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> 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       Log On


   

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) About Me  (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 (x%5B1%5D,y%5B1%5D) be that first point (ie x%5B1%5D=2 and y%5B1%5D=120)


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 (x%5B2%5D,y%5B2%5D) be the second point (ie x%5B2%5D=3 and y%5B2%5D=60)


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 (2,120) and (3,60)

m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula (note: (x%5B1%5D,y%5B1%5D) is the first point (2,120) and (x%5B2%5D,y%5B2%5D) is the second point (3,60))

m=%2860-120%29%2F%283-2%29 Plug in y%5B2%5D=60,y%5B1%5D=120,x%5B2%5D=3,x%5B1%5D=2 (these are the coordinates of given points)

m=+-60%2F1 Subtract the terms in the numerator 60-120 to get -60. Subtract the terms in the denominator 3-2 to get 1


So the slope is
m=-60




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



------Point-Slope Formula------
y-y%5B1%5D=m%28x-x%5B1%5D%29 where m is the slope, and (x%5B1%5D,y%5B1%5D) is one of the given points

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

y-120=%28-60%29%28x-2%29 Plug in m=-60, x%5B1%5D=2, and y%5B1%5D=120 (these values are given)


y-120=-60x%2B%28-60%29%28-2%29 Distribute -60

y-120=-60x%2B120 Multiply -60 and -2 to get 120


y=-60x%2B120%2B120 Add 120 to both sides to isolate y

y=-60x%2B240 Combine like terms 120 and 120 to get 240
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Answer:


So the equation of the line which goes through the points (2,120) and (3,60) is:y=-60x%2B240

The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=-60 and the y-intercept is b=240

Notice if we graph the equation y=-60x%2B240 and plot the points (2,120) and (3,60), we get this: (note: if you need help with graphing, check out this solver)

Graph of y=-60x%2B240 through the points (2,120) and (3,60)

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


Since the equation is y=-60x%2B240 the function is C%28p%29=-60p%2B240 where C%28p%29 is the cost (relative to a given price) and p is the price