Question 902401

First let's find the slope of the line through the points *[Tex \LARGE \left(3,1\right)] and *[Tex \LARGE \left(-2,-9\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(3,1\right)]. So this means that {{{x[1]=3}}} and {{{y[1]=1}}}.

Also, *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(-2,-9\right)].  So this means that {{{x[2]=-2}}} and {{{y[2]=-9}}}.



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



{{{m=(-9-1)/(-2-3)}}} Plug in {{{y[2]=-9}}}, {{{y[1]=1}}}, {{{x[2]=-2}}}, and {{{x[1]=3}}}



{{{m=(-10)/(-2-3)}}} Subtract {{{1}}} from {{{-9}}} to get {{{-10}}}



{{{m=(-10)/(-5)}}} Subtract {{{3}}} from {{{-2}}} to get {{{-5}}}



{{{m=2}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(3,1\right)] and *[Tex \LARGE \left(-2,-9\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-1=2(x-3)}}} Plug in {{{m=2}}}, {{{x[1]=3}}}, and {{{y[1]=1}}}



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



{{{y-1=2x-6}}} Multiply



{{{y=2x-6+1}}} Add 1 to both sides. 



{{{y=2x-5}}} Combine like terms. 



So the equation that goes through the points *[Tex \LARGE \left(3,1\right)] and *[Tex \LARGE \left(-2,-9\right)] is {{{y=2x-5}}}



 Notice how the graph of {{{y=2x-5}}} goes through the points *[Tex \LARGE \left(3,1\right)] and *[Tex \LARGE \left(-2,-9\right)]. So this visually verifies our answer.

 {{{drawing( 500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,0,2x-5),
 circle(3,1,0.08),
 circle(3,1,0.10),
 circle(3,1,0.12),
 circle(-2,-9,0.08),
 circle(-2,-9,0.10),
 circle(-2,-9,0.12)
 )}}} Graph of {{{y=2x-5}}} through the points *[Tex \LARGE \left(3,1\right)] and *[Tex \LARGE \left(-2,-9\right)]

 
y-intercept is (0,-5)
x-intercept is (2.5, 0)