Question 567787


First let's find the slope of the line through the points *[Tex \LARGE \left(2,5\right)] and *[Tex \LARGE \left(6,7\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(2,5\right)]. So this means that {{{x[1]=2}}} and {{{y[1]=5}}}.

Also, *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(6,7\right)].  So this means that {{{x[2]=6}}} and {{{y[2]=7}}}.



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



{{{m=(7-5)/(6-2)}}} Plug in {{{y[2]=7}}}, {{{y[1]=5}}}, {{{x[2]=6}}}, and {{{x[1]=2}}}



{{{m=(2)/(6-2)}}} Subtract {{{5}}} from {{{7}}} to get {{{2}}}



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



{{{m=1/2}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(2,5\right)] and *[Tex \LARGE \left(6,7\right)] is {{{m=1/2}}}



Now let's use the point slope formula:



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



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



{{{y-5=(1/2)x+(1/2)(-2)}}} Distribute



{{{y-5=(1/2)x-1}}} Multiply



{{{y=(1/2)x-1+5}}} Add 5 to both sides. 



{{{y=(1/2)x+4}}} Combine like terms. 



{{{y=(1/2)x+4}}} Simplify



So the equation that goes through the points *[Tex \LARGE \left(2,5\right)] and *[Tex \LARGE \left(6,7\right)] is {{{y=(1/2)x+4}}}



 Notice how the graph of {{{y=(1/2)x+4}}} goes through the points *[Tex \LARGE \left(2,5\right)] and *[Tex \LARGE \left(6,7\right)]. So this visually verifies our answer.

 {{{drawing( 500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,(1/2)x+4),
 circle(2,5,0.08),
 circle(2,5,0.10),
 circle(2,5,0.12),
 circle(6,7,0.08),
 circle(6,7,0.10),
 circle(6,7,0.12)
 )}}} Graph of {{{y=(1/2)x+4}}} through the points *[Tex \LARGE \left(2,5\right)] and *[Tex \LARGE \left(6,7\right)]