Question 121745
Note: I'm going to use "y" instead of "p" 


Since the price is p=$20, this means y=20 when x=42. So we have the first point (42,20).



Also since the price is p=$10, this means y=10 when x=52. So we have the second point (10,52).


So let's find the equation of the line through the points (42,20) and (10,52):


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First lets find the slope through the points ({{{42}}},{{{20}}}) and ({{{10}}},{{{52}}})


{{{m=(y[2]-y[1])/(x[2]-x[1])}}} Start with the slope formula (note: *[Tex \Large \left(x_{1},y_{1}\right)] is the first point ({{{42}}},{{{20}}}) and  *[Tex \Large \left(x_{2},y_{2}\right)] is the second point ({{{10}}},{{{52}}}))


{{{m=(52-20)/(10-42)}}} Plug in {{{y[2]=52}}},{{{y[1]=20}}},{{{x[2]=10}}},{{{x[1]=42}}}  (these are the coordinates of given points)


{{{m= 32/-32}}} Subtract the terms in the numerator {{{52-20}}} to get {{{32}}}.  Subtract the terms in the denominator {{{10-42}}} to get {{{-32}}}

  


{{{m=-1}}} Reduce

  

So the slope is

{{{m=-1}}}


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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 *[Tex \Large \left(\textrm{x_{1},y_{1}}\right)] is one of the given points


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


{{{y-20=(-1)(x-42)}}} Plug in {{{m=-1}}}, {{{x[1]=42}}}, and {{{y[1]=20}}} (these values are given)



{{{y-20=-x+(-1)(-42)}}} Distribute {{{-1}}}


{{{y-20=-x+42}}} Multiply {{{-1}}} and {{{-42}}} to get {{{42}}}


{{{y=-x+42+20}}} Add {{{20}}} to  both sides to isolate y


{{{y=-x+62}}} Combine like terms {{{42}}} and {{{20}}} to get {{{62}}} 




So the equation of the line which goes through the points ({{{42}}},{{{20}}}) and ({{{10}}},{{{52}}})  is:{{{y=-x+62}}}


{{{p=-x+62}}} Now replace y with p 


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Answer:


So the demand equation is:

{{{p=-x+62}}}