Question 205953

First let's find the slope of the line through the points *[Tex \LARGE \left(5,17\right)] and *[Tex \LARGE \left(9,13\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(5,17\right)]. So this means that {{{x[1]=5}}} and {{{y[1]=17}}}.

Also, *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(9,13\right)].  So this means that {{{x[2]=9}}} and {{{y[2]=13}}}.



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



{{{m=(13-17)/(9-5)}}} Plug in {{{y[2]=13}}}, {{{y[1]=17}}}, {{{x[2]=9}}}, and {{{x[1]=5}}}



{{{m=(-4)/(9-5)}}} Subtract {{{17}}} from {{{13}}} to get {{{-4}}}



{{{m=(-4)/(4)}}} Subtract {{{5}}} from {{{9}}} to get {{{4}}}



{{{m=-1}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(5,17\right)] and *[Tex \LARGE \left(9,13\right)] is {{{m=-1}}}



Now let's use the point slope formula:



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



{{{y-17=-1(x-5)}}} Plug in {{{m=-1}}}, {{{x[1]=5}}}, and {{{y[1]=17}}}



{{{y-17=-1x+-1(-5)}}} Distribute



{{{y-17=-1x+5}}} Multiply



{{{y=-1x+5+17}}} Add 17 to both sides. 



{{{y=-1x+22}}} Combine like terms. 



{{{y=-x+22}}} Simplify



So the equation that goes through the points *[Tex \LARGE \left(5,17\right)] and *[Tex \LARGE \left(9,13\right)] is {{{y=-x+22}}}