Question 559807


First let's find the slope of the line through the points *[Tex \LARGE \left(6,10\right)] and *[Tex \LARGE \left(4,1\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(6,10\right)]. So this means that {{{x[1]=6}}} and {{{y[1]=10}}}.

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



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



{{{m=(1-10)/(4-6)}}} Plug in {{{y[2]=1}}}, {{{y[1]=10}}}, {{{x[2]=4}}}, and {{{x[1]=6}}}



{{{m=(-9)/(4-6)}}} Subtract {{{10}}} from {{{1}}} to get {{{-9}}}



{{{m=(-9)/(-2)}}} Subtract {{{6}}} from {{{4}}} to get {{{-2}}}



{{{m=9/2}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(6,10\right)] and *[Tex \LARGE \left(4,1\right)] is {{{m=9/2}}}



Now let's use the point slope formula:



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



{{{y-10=(9/2)(x-6)}}} Plug in {{{m=9/2}}}, {{{x[1]=6}}}, and {{{y[1]=10}}}



{{{y-10=(9/2)x+(9/2)(-6)}}} Distribute



{{{y-10=(9/2)x-27}}} Multiply



{{{y=(9/2)x-27+10}}} Add 10 to both sides. 



{{{y=(9/2)x-17}}} Combine like terms. 



{{{y=(9/2)x-17}}} Simplify



So the equation that goes through the points *[Tex \LARGE \left(6,10\right)] and *[Tex \LARGE \left(4,1\right)] is {{{y=(9/2)x-17}}}



 Notice how the graph of {{{y=(9/2)x-17}}} goes through the points *[Tex \LARGE \left(6,10\right)] and *[Tex \LARGE \left(4,1\right)]. So this visually verifies our answer.

 {{{drawing( 500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,(9/2)x-17),
 circle(6,10,0.08),
 circle(6,10,0.10),
 circle(6,10,0.12),
 circle(4,1,0.08),
 circle(4,1,0.10),
 circle(4,1,0.12)
 )}}} Graph of {{{y=(9/2)x-17}}} through the points *[Tex \LARGE \left(6,10\right)] and *[Tex \LARGE \left(4,1\right)]

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