Question 350954

First let's find the slope of the line through the points *[Tex \LARGE \left(-5,6\right)] and *[Tex \LARGE \left(0,-2\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(-5,6\right)]. So this means that {{{x[1]=-5}}} and {{{y[1]=6}}}.

Also, *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(0,-2\right)].  So this means that {{{x[2]=0}}} and {{{y[2]=-2}}}.



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



{{{m=(-2-6)/(0--5)}}} Plug in {{{y[2]=-2}}}, {{{y[1]=6}}}, {{{x[2]=0}}}, and {{{x[1]=-5}}}



{{{m=(-8)/(0--5)}}} Subtract {{{6}}} from {{{-2}}} to get {{{-8}}}



{{{m=(-8)/(5)}}} Subtract {{{-5}}} from {{{0}}} to get {{{5}}}



So the slope of the line that goes through the points *[Tex \LARGE \left(-5,6\right)] and *[Tex \LARGE \left(0,-2\right)] is {{{m=-8/5}}}



Now let's use the point slope formula:



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



{{{y-6=(-8/5)(x--5)}}} Plug in {{{m=-8/5}}}, {{{x[1]=-5}}}, and {{{y[1]=6}}}



{{{y-6=(-8/5)(x+5)}}} Rewrite {{{x--5}}} as {{{x+5}}}



{{{y-6=(-8/5)x+(-8/5)(5)}}} Distribute



{{{y-6=(-8/5)x-8}}} Multiply



{{{y=(-8/5)x-8+6}}} Add 6 to both sides. 



{{{y=(-8/5)x-2}}} Combine like terms. 



So the equation that goes through the points *[Tex \LARGE \left(-5,6\right)] and *[Tex \LARGE \left(0,-2\right)] is {{{y=(-8/5)x-2}}}



 Notice how the graph of {{{y=(-8/5)x-2}}} goes through the points *[Tex \LARGE \left(-5,6\right)] and *[Tex \LARGE \left(0,-2\right)]. So this visually verifies our answer.

 {{{drawing( 500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,(-8/5)x-2),
 circle(-5,6,0.08),
 circle(-5,6,0.10),
 circle(-5,6,0.12),
 circle(0,-2,0.08),
 circle(0,-2,0.10),
 circle(0,-2,0.12)
 )}}} Graph of {{{y=(-8/5)x-2}}} through the points *[Tex \LARGE \left(-5,6\right)] and *[Tex \LARGE \left(0,-2\right)]

 


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