Question 451297


First let's find the slope of the line through the points *[Tex \LARGE \left(-5,8\right)] and *[Tex \LARGE \left(-3,-8\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(-5,8\right)]. So this means that {{{x[1]=-5}}} and {{{y[1]=8}}}.

Also, *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(-3,-8\right)].  So this means that {{{x[2]=-3}}} and {{{y[2]=-8}}}.



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



{{{m=(-8-8)/(-3--5)}}} Plug in {{{y[2]=-8}}}, {{{y[1]=8}}}, {{{x[2]=-3}}}, and {{{x[1]=-5}}}



{{{m=(-16)/(-3--5)}}} Subtract {{{8}}} from {{{-8}}} to get {{{-16}}}



{{{m=(-16)/(2)}}} Subtract {{{-5}}} from {{{-3}}} to get {{{2}}}



{{{m=-8}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(-5,8\right)] and *[Tex \LARGE \left(-3,-8\right)] is {{{m=-8}}}



Now let's use the point slope formula:



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



{{{y-8=-8(x--5)}}} Plug in {{{m=-8}}}, {{{x[1]=-5}}}, and {{{y[1]=8}}}



{{{y-8=-8(x+5)}}} Rewrite {{{x--5}}} as {{{x+5}}}



{{{y-8=-8x+-8(5)}}} Distribute



{{{y-8=-8x-40}}} Multiply



{{{y=-8x-40+8}}} Add 8 to both sides. 



{{{y=-8x-32}}} Combine like terms. 



{{{y=-8x-32}}} Simplify



So the equation that goes through the points *[Tex \LARGE \left(-5,8\right)] and *[Tex \LARGE \left(-3,-8\right)] is {{{y=-8x-32}}}



 Notice how the graph of {{{y=-8x-32}}} goes through the points *[Tex \LARGE \left(-5,8\right)] and *[Tex \LARGE \left(-3,-8\right)]. So this visually verifies our answer.

 {{{drawing( 500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,-8x-32),
 circle(-5,8,0.08),
 circle(-5,8,0.10),
 circle(-5,8,0.12),
 circle(-3,-8,0.08),
 circle(-3,-8,0.10),
 circle(-3,-8,0.12)
 )}}} Graph of {{{y=-8x-32}}} through the points *[Tex \LARGE \left(-5,8\right)] and *[Tex \LARGE \left(-3,-8\right)]