Question 350964


First let's find the slope of the line through the points *[Tex \LARGE \left(1,6\right)] and *[Tex \LARGE \left(-1,2\right)]



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

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



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



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



{{{m=(-4)/(-1-1)}}} Subtract {{{6}}} from {{{2}}} to get {{{-4}}}



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



{{{m=2}}} Reduce



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



Now let's use the point slope formula:



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



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



{{{y-6=2x+2(-1)}}} Distribute



{{{y-6=2x-2}}} Multiply



{{{y=2x-2+6}}} Add 6 to both sides. 



{{{y=2x+4}}} Combine like terms. 



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



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

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

 

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