Question 195588

First let's find the slope of the line through the points *[Tex \LARGE \left(3,5\right)] and *[Tex \LARGE \left(5,12\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(3,5\right)] and *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(5,12\right)].



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



{{{m=(12-5)/(5-3)}}} Plug in {{{y[2]=12}}}, {{{y[1]=5}}}, {{{x[2]=5}}}, and {{{x[1]=3}}}



{{{m=(7)/(5-3)}}} Subtract {{{5}}} from {{{12}}} to get {{{7}}}



{{{m=(7)/(2)}}} Subtract {{{3}}} from {{{5}}} to get {{{2}}}



So the slope of the line that goes through the points *[Tex \LARGE \left(3,5\right)] and *[Tex \LARGE \left(5,12\right)] is {{{m=7/2}}}



Now let's use the point slope formula:



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



{{{y-5=(7/2)(x-3)}}} Plug in {{{m=7/2}}}, {{{x[1]=3}}}, and {{{y[1]=5}}}



{{{y-5=(7/2)x+(7/2)(-3)}}} Distribute



{{{y-5=(7/2)x-21/2}}} Multiply



{{{y=(7/2)x-21/2+5}}} Add 5 to both sides. 



{{{y=(7/2)x-11/2}}} Combine like terms. 



So the equation that goes through the points *[Tex \LARGE \left(3,5\right)] and *[Tex \LARGE \left(5,12\right)] is {{{y=(7/2)x-11/2}}}