Question 544699


First let's find the slope of the line through the points *[Tex \LARGE \left(2,1\right)] and *[Tex \LARGE \left(3,5\right)]



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

Also, *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(3,5\right)].  So this means that {{{x[2]=3}}} and {{{y[2]=5}}}.



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



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



{{{m=(4)/(3-2)}}} Subtract {{{1}}} from {{{5}}} to get {{{4}}}



{{{m=(4)/(1)}}} Subtract {{{2}}} from {{{3}}} to get {{{1}}}



{{{m=4}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(2,1\right)] and *[Tex \LARGE \left(3,5\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-1=4(x-2)}}} Plug in {{{m=4}}}, {{{x[1]=2}}}, and {{{y[1]=1}}}



{{{y-1=4x+4(-2)}}} Distribute



{{{y-1=4x-8}}} Multiply



{{{y=4x-8+1}}} Add 1 to both sides. 



{{{y=4x-7}}} Combine like terms. 



So the equation that goes through the points *[Tex \LARGE \left(2,1\right)] and *[Tex \LARGE \left(3,5\right)] is {{{y=4x-7}}}



 Notice how the graph of {{{y=4x-7}}} goes through the points *[Tex \LARGE \left(2,1\right)] and *[Tex \LARGE \left(3,5\right)]. So this visually verifies our answer.

 {{{drawing( 500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,4x-7),
 circle(2,1,0.08),
 circle(2,1,0.10),
 circle(2,1,0.12),
 circle(3,5,0.08),
 circle(3,5,0.10),
 circle(3,5,0.12)
 )}}} Graph of {{{y=4x-7}}} through the points *[Tex \LARGE \left(2,1\right)] and *[Tex \LARGE \left(3,5\right)]