Question 274739
First let's find the slope of the line through the points *[Tex \LARGE \left(-3,-1\right)] and *[Tex \LARGE \left(-9,-6\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(-9,-6\right)].  So this means that {{{x[2]=-9}}} and {{{y[2]=-6}}}.

{{{m=(-6--1)/(-9--3)}}} Plug in {{{y[2]=-6}}}, {{{y[1]=-1}}}, {{{x[2]=-9}}}, and {{{x[1]=-3}}}

{{{m=(-5)/(-9--3)}}} Subtract {{{-1}}} from {{{-6}}} to get {{{-5}}}

{{{m=(-5)/(-6)}}} Subtract {{{-3}}} from {{{-9}}} to get {{{-6}}}

{{{m=5/6}}} Reduce

So the slope of the line that goes through the points *[Tex \LARGE \left(-3,-1\right)] and *[Tex \LARGE \left(-9,-6\right)] is {{{m=5/6}}}

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Now remember that the general slope intercept equation is {{{y=mx+b}}} where 'm' is the slope of the line and 'b' is the y-intercept. We can use this general equation to find the equation of the line.

Since the line goes through the point (-3,-1), this means that x=-3 and y=-1. In addition, we know that the slope is {{{m=5/6}}}. So we can use these values to solve for 'b'.

{{{-1=(5/6)(-3)+b}}} Plug in {{{x=-3}}}, {{{y=-1}}} and {{{m=5/6}}}

{{{-1=-15/6+b}}} Multiply.

{{{-1=-5/2+b}}} Reduce.

{{{-1+5/2=b}}} Add {{{5/2}}} to both sides to isolate 'b'

{{{3/2=b}}} Combine like terms.

So the value of 'b' is {{{b=3/2}}}

Since {{{m=5/6}}} and {{{b=3/2}}}, we can plug these values into {{{y=mx+b}}} to get {{{y=(5/6)x+3/2}}}

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So the equation of the line in slope-intercept form through the points (-3,-1) and (-9,-6) is {{{y=(5/6)x+3/2}}}