Question 117384
First lets find the slope through the points ({{{2}}},{{{1}}}) and ({{{1}}},{{{2}}})


{{{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 ({{{2}}},{{{1}}}) and  *[Tex \Large \left(x_{2},y_{2}\right)] is the second point ({{{1}}},{{{2}}}))


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


{{{m= 1/-1}}} Subtract the terms in the numerator {{{2-1}}} to get {{{1}}}.  Subtract the terms in the denominator {{{1-2}}} to get {{{-1}}}

  


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



{{{y-1=-x+(-1)(-2)}}} Distribute {{{-1}}}


{{{y-1=-x+2}}} Multiply {{{-1}}} and {{{-2}}} to get {{{2}}}


{{{y=-x+2+1}}} Add {{{1}}} to  both sides to isolate y


{{{y=-x+3}}} Combine like terms {{{2}}} and {{{1}}} to get {{{3}}} 

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Answer:



So the equation of the line which goes through the points ({{{2}}},{{{1}}}) and ({{{1}}},{{{2}}})  is:{{{y=-x+3}}}


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=3}}}


Notice if we graph the equation {{{y=-x+3}}} and plot the points ({{{2}}},{{{1}}}) and ({{{1}}},{{{2}}}),  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, -7.5, 10.5, -7.5, 10.5,
graph(500, 500, -7.5, 10.5, -7.5, 10.5,(-1)x+3),
circle(2,1,0.12),
circle(2,1,0.12+0.03),
circle(1,2,0.12),
circle(1,2,0.12+0.03)
) }}} Graph of {{{y=-x+3}}} through the points ({{{2}}},{{{1}}}) and ({{{1}}},{{{2}}})


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