Question 103953
Think of c as y to get one pair of values: x=12 million and y=18 million
and another pair: x=18 million and y=24 million



So we basically can think of these pairs as points on a coordinate system 

(12,18) and (18,24) (notice I've taken out the "millions". This will simplify things if we remember everything is in millions and not write it every time)



So let's find the equation of the line through the points (12,18) and (18,24) 




First lets find the slope through the points ({{{12}}},{{{18}}}) and ({{{18}}},{{{24}}})


{{{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 ({{{12}}},{{{18}}}) and  *[Tex \Large \left(x_{2},y_{2}\right)] is the second point ({{{18}}},{{{24}}}))


{{{m=(24-18)/(18-12)}}} Plug in {{{y[2]=24}}},{{{y[1]=18}}},{{{x[2]=18}}},{{{x[1]=12}}}  (these are the coordinates of given points)


{{{m= 6/6}}} Subtract the terms in the numerator {{{24-18}}} to get {{{6}}}.  Subtract the terms in the denominator {{{18-12}}} to get {{{6}}}

  


{{{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-18=(1)(x-12)}}} Plug in {{{m=1}}}, {{{x[1]=12}}}, and {{{y[1]=18}}} (these values are given)



{{{y-18=1x+(1)(-12)}}} Distribute {{{1}}}


{{{y-18=1x-12}}} Multiply {{{1}}} and {{{-12}}} to get {{{-12}}}


{{{y=1x-12+18}}} Add {{{18}}} to  both sides to isolate y


{{{y=1x+6}}} Combine like terms {{{-12}}} and {{{18}}} to get {{{6}}} 

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Answer:



So the equation of the line which goes through the points ({{{12}}},{{{18}}}) and ({{{18}}},{{{24}}})  is:{{{y=1x+6}}}


The equation is now in {{{y=mx+b}}} form (which is slope-intercept form) where the slope is {{{m=1}}} and the y-intercept is {{{b=6}}}


Notice if we graph the equation {{{y=1x+6}}} and plot the points ({{{12}}},{{{18}}}) and ({{{18}}},{{{24}}}),  we get this: (note: if you need help with graphing, check out this <a href=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver<a>)


{{{drawing(500, 500, -6, 24, -12, 30,
graph(500, 500, -6, 24, -12, 30,(1)x+6),
circle(12,18,0.12),
circle(12,18,0.12+0.03),
circle(18,24,0.12),
circle(18,24,0.12+0.03)
) }}} Graph of {{{y=1x+6}}} through the points ({{{12}}},{{{18}}}) and ({{{18}}},{{{24}}})


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



So to make a function out of {{{y=x+6}}}, simply replace y with {{{C(x)}}} to get 



{{{C(x)=x+6}}}