Question 585857
I'll do the first one to get you started.




First let's find the slope of the line through the points *[Tex \LARGE \left(-1,2\right)] and *[Tex \LARGE \left(1,-4\right)]



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

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



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



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



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



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



{{{m=-3}}} Reduce



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



{{{y-2=-3(x+1)}}} Rewrite {{{x--1}}} as {{{x+1}}}



{{{y-2=-3x+-3(1)}}} Distribute



{{{y-2=-3x-3}}} Multiply



{{{y=-3x-3+2}}} Add 2 to both sides. 



{{{y=-3x-1}}} Combine like terms. 



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



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

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

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