Question 173333
Let x=elevation (in feet) and y=temperature (in Fahrenheit)



So at 6000 ft, the temperature is 76F, this means that {{{x=6000}}} and {{{y=76}}} 


So we have one point (6000,76)





So at 12000 ft, the temperature is 49F, this means that {{{x=12000}}} and {{{y=49}}} 


So we have one point (12000,49)



So let's find the equation of the line through the two points (6000,76) and (12000,49)



First let's find the slope of the line through the points *[Tex \LARGE \left(6000,76\right)] and *[Tex \LARGE \left(12000,49\right)]



{{{m=(y[2]-y[1])/(x[2]-x[1])}}} Start with the slope formula.



{{{m=(49-76)/(12000-6000)}}} Plug in {{{y[2]=49}}}, {{{y[1]=76}}}, {{{x[2]=12000}}}, and {{{x[1]=6000}}}



{{{m=(-27)/(12000-6000)}}} Subtract {{{76}}} from {{{49}}} to get {{{-27}}}



{{{m=(-27)/(6000)}}} Subtract {{{6000}}} from {{{12000}}} to get {{{6000}}}



{{{m=-9/2000}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(6000,76\right)] and *[Tex \LARGE \left(12000,49\right)] is {{{m=-9/2000}}}



Now let's use the point slope formula:



{{{y-y[1]=m(x-x[1])}}} Start with the point slope formula



{{{y-76=(-9/2000)(x-6000)}}} Plug in {{{m=-9/2000}}}, {{{x[1]=6000}}}, and {{{y[1]=76}}}



{{{y-76=(-9/2000)x+(-9/2000)(-6000)}}} Distribute



{{{y-76=(-9/2000)x+27}}} Multiply



{{{y=(-9/2000)x+27+76}}} Add 76 to both sides. 



{{{y=(-9/2000)x+103}}} Combine like terms. 



{{{y=(-9/2000)x+103}}} Simplify



So the equation that goes through the points *[Tex \LARGE \left(6000,76\right)] and *[Tex \LARGE \left(12000,49\right)] is {{{y=(-9/2000)x+103}}}




So the equation that represents the temperature T at elevation E is



{{{T=(-9/2000)E+103}}}



Note: just replace "x" and "y" with T and E