Question 146876
I'll do the first one to get you started


# 1




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



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



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



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



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



{{{m=1}}} Reduce



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



Now let's use the point slope formula:



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



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



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



{{{y-1=1x-3}}} Multiply



{{{y=1x-3+1}}} Add 1 to both sides. 



{{{y=1x-2}}} Combine like terms. 



{{{y=x-2}}} Simplify



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



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

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