Question 107873
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[1]}}},{{{y[1]}}}) be that first point (ie {{{x[1]=2}}} and {{{y[1]=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[2]}}},{{{y[2]}}}) be the second point (ie {{{x[2]=3}}} and {{{y[2]=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=(y[2]-y[1])/(x[2]-x[1])}}} Start with the slope formula (note: ({{{x[1]}}},{{{y[1]}}}) is the first point ({{{2}}},{{{120}}}) and ({{{x[2]}}},{{{y[2]}}}) is the second point ({{{3}}},{{{60}}}))


{{{m=(60-120)/(3-2)}}} Plug in {{{y[2]=60}}},{{{y[1]=120}}},{{{x[2]=3}}},{{{x[1]=2}}} (these are the coordinates of given points)


{{{m= -60/1}}} 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[1]=m(x-x[1])}}} where {{{m}}} is the slope, and ({{{x[1]}}},{{{y[1]}}}) is one of the given points


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


{{{y-120=(-60)(x-2)}}} Plug in {{{m=-60}}}, {{{x[1]=2}}}, and {{{y[1]=120}}} (these values are given)



{{{y-120=-60x+(-60)(-2)}}} Distribute {{{-60}}}


{{{y-120=-60x+120}}} Multiply {{{-60}}} and {{{-2}}} to get {{{120}}}



{{{y=-60x+120+120}}} Add {{{120}}} to both sides to isolate y


{{{y=-60x+240}}} 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+240}}}


The equation is now in {{{y=mx+b}}} 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+240}}} and plot the points ({{{2}}},{{{120}}}) and ({{{3}}},{{{60}}}), we get this: (note: if you need help with graphing, check out this solver)


{{{drawing(500, 500, -6.5, 11.5, 45, 150,
graph(500, 500, -6.5, 11.5, 45, 150,(-60)x+240),
circle(2,120,0.12),
circle(2,120,0.12+0.03),
circle(3,60,0.12),
circle(3,60,0.12+0.03)
) }}} Graph of {{{y=-60x+240}}} 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+240}}} the function is {{{C(p)=-60p+240}}} where {{{C(p)}}} is the cost (relative to a given price) and p is the price