Question 220792
First let's find the slope of the line through the points *[Tex \LARGE \left(2,4\right)] and *[Tex \LARGE \left(5,16\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(2,4\right)]. So this means that {{{x[1]=2}}} and {{{y[1]=4}}}.

Also, *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(5,16\right)].  So this means that {{{x[2]=5}}} and {{{y[2]=16}}}.



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



{{{m=(16-4)/(5-2)}}} Plug in {{{y[2]=16}}}, {{{y[1]=4}}}, {{{x[2]=5}}}, and {{{x[1]=2}}}



{{{m=(12)/(5-2)}}} Subtract {{{4}}} from {{{16}}} to get {{{12}}}



{{{m=(12)/(3)}}} Subtract {{{2}}} from {{{5}}} to get {{{3}}}



{{{m=4}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(2,4\right)] and *[Tex \LARGE \left(5,16\right)] is {{{m=4}}}



Now let's use the point slope formula:



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



{{{y-4=4(x-2)}}} Plug in {{{m=4}}}, {{{x[1]=2}}}, and {{{y[1]=4}}}



{{{y-4=4x+4(-2)}}} Distribute



{{{y-4=4x-8}}} Multiply



{{{y=4x-8+4}}} Add 4 to both sides. 



{{{y=4x-4}}} Combine like terms. 



So the equation that goes through the points *[Tex \LARGE \left(2,4\right)] and *[Tex \LARGE \left(5,16\right)] is {{{y=4x-4}}}



 Notice how the graph of {{{y=4x-4}}} goes through the points *[Tex \LARGE \left(2,4\right)] and *[Tex \LARGE \left(5,16\right)]. So this visually verifies our answer.

 {{{drawing( 500, 500, -10, 10, -5, 18,
 graph( 500, 500, -10, 10, -5, 18,4x-4),
 circle(2,4,0.08),
 circle(2,4,0.10),
 circle(2,4,0.12),
 circle(5,16,0.08),
 circle(5,16,0.10),
 circle(5,16,0.12)
 )}}} Graph of {{{y=4x-4}}} through the points *[Tex \LARGE \left(2,4\right)] and *[Tex \LARGE \left(5,16\right)]