Question 281589

First let's find the slope of the line through the points *[Tex \LARGE \left(2,0\right)] and *[Tex \LARGE \left(4,6\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(2,0\right)]. So this means that {{{x[1]=2}}} and {{{y[1]=0}}}.

Also, *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(4,6\right)].  So this means that {{{x[2]=4}}} and {{{y[2]=6}}}.



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



{{{m=(6-0)/(4-2)}}} Plug in {{{y[2]=6}}}, {{{y[1]=0}}}, {{{x[2]=4}}}, and {{{x[1]=2}}}



{{{m=(6)/(4-2)}}} Subtract {{{0}}} from {{{6}}} to get {{{6}}}



{{{m=(6)/(2)}}} Subtract {{{2}}} from {{{4}}} to get {{{2}}}



{{{m=3}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(2,0\right)] and *[Tex \LARGE \left(4,6\right)] is {{{m=3}}}



Now let's use the point slope formula:



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



{{{y-0=3(x-2)}}} Plug in {{{m=3}}}, {{{x[1]=2}}}, and {{{y[1]=0}}}



{{{y-0=3x+3(-2)}}} Distribute



{{{y-0=3x-6}}} Multiply



{{{y=3x-6}}} Simplify




So the equation that goes through the points *[Tex \LARGE \left(2,0\right)] and *[Tex \LARGE \left(4,6\right)] is {{{y=3x-6}}}



 Notice how the graph of {{{y=3x-6}}} goes through the points *[Tex \LARGE \left(2,0\right)] and *[Tex \LARGE \left(4,6\right)]. So this visually verifies our answer.

 {{{drawing( 500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,3x-6),
 circle(2,0,0.08),
 circle(2,0,0.10),
 circle(2,0,0.12),
 circle(4,6,0.08),
 circle(4,6,0.10),
 circle(4,6,0.12)
 )}}} Graph of {{{y=3x-6}}} through the points *[Tex \LARGE \left(2,0\right)] and *[Tex \LARGE \left(4,6\right)]