Question 236559

First let's find the slope of the line through the points *[Tex \LARGE \left(-1,-3\right)] and *[Tex \LARGE \left(-5,-6\right)]



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

Also, *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(-5,-6\right)].  So this means that {{{x[2]=-5}}} and {{{y[2]=-6}}}.



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



{{{m=(-6--3)/(-5--1)}}} Plug in {{{y[2]=-6}}}, {{{y[1]=-3}}}, {{{x[2]=-5}}}, and {{{x[1]=-1}}}



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



{{{m=(-3)/(-4)}}} Subtract {{{-1}}} from {{{-5}}} to get {{{-4}}}



{{{m=3/4}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(-1,-3\right)] and *[Tex \LARGE \left(-5,-6\right)] is {{{m=3/4}}}



Now let's use the point slope formula:



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



{{{y--3=(3/4)(x--1)}}} Plug in {{{m=3/4}}}, {{{x[1]=-1}}}, and {{{y[1]=-3}}}



{{{y--3=(3/4)(x+1)}}} Rewrite {{{x--1}}} as {{{x+1}}}



{{{y+3=(3/4)(x+1)}}} Rewrite {{{y--3}}} as {{{y+3}}}



{{{y+3=(3/4)x+(3/4)(1)}}} Distribute



{{{y+3=(3/4)x+3/4}}} Multiply



{{{y=(3/4)x+3/4-3}}} Subtract 3 from both sides. 



{{{y=(3/4)x-9/4}}} Combine like terms. note: If you need help with fractions, check out this <a href="http://www.algebra.com/algebra/homework/NumericFractions/fractions-solver.solver">solver</a>.




So the equation that goes through the points *[Tex \LARGE \left(-1,-3\right)] and *[Tex \LARGE \left(-5,-6\right)] is {{{y=(3/4)x-9/4}}}



 Notice how the graph of {{{y=(3/4)x-9/4}}} goes through the points *[Tex \LARGE \left(-1,-3\right)] and *[Tex \LARGE \left(-5,-6\right)]. So this visually verifies our answer.

 {{{drawing( 500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,(3/4)x-9/4),
 circle(-1,-3,0.08),
 circle(-1,-3,0.10),
 circle(-1,-3,0.12),
 circle(-5,-6,0.08),
 circle(-5,-6,0.10),
 circle(-5,-6,0.12)
 )}}} Graph of {{{y=(3/4)x-9/4}}} through the points *[Tex \LARGE \left(-1,-3\right)] and *[Tex \LARGE \left(-5,-6\right)]