Question 255379

First let's find the slope of the line through the points *[Tex \LARGE \left(0,-4\right)] and *[Tex \LARGE \left(-6,7\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(0,-4\right)]. So this means that {{{x[1]=0}}} and {{{y[1]=-4}}}.

Also, *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(-6,7\right)].  So this means that {{{x[2]=-6}}} and {{{y[2]=7}}}.



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



{{{m=(7--4)/(-6-0)}}} Plug in {{{y[2]=7}}}, {{{y[1]=-4}}}, {{{x[2]=-6}}}, and {{{x[1]=0}}}



{{{m=(11)/(-6-0)}}} Subtract {{{-4}}} from {{{7}}} to get {{{11}}}



{{{m=(11)/(-6)}}} Subtract {{{0}}} from {{{-6}}} to get {{{-6}}}



{{{m=-11/6}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(0,-4\right)] and *[Tex \LARGE \left(-6,7\right)] is {{{m=-11/6}}}



Now let's use the point slope formula:



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



{{{y--4=(-11/6)(x-0)}}} Plug in {{{m=-11/6}}}, {{{x[1]=0}}}, and {{{y[1]=-4}}}



{{{y+4=(-11/6)(x-0)}}} Rewrite {{{y--4}}} as {{{y+4}}}



{{{y+4=(-11/6)x+(-11/6)(0)}}} Distribute



{{{y+4=(-11/6)x+0}}} Multiply



{{{y=(-11/6)x+0-4}}} Subtract 4 from both sides. 



{{{y=(-11/6)x-4}}} Combine like terms. 



So the equation that goes through the points *[Tex \LARGE \left(0,-4\right)] and *[Tex \LARGE \left(-6,7\right)] is {{{y=(-11/6)x-4}}}



 Notice how the graph of {{{y=(-11/6)x-4}}} goes through the points *[Tex \LARGE \left(0,-4\right)] and *[Tex \LARGE \left(-6,7\right)]. So this visually verifies our answer.

 {{{drawing( 500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,(-11/6)x-4),
 circle(0,-4,0.08),
 circle(0,-4,0.10),
 circle(0,-4,0.12),
 circle(-6,7,0.08),
 circle(-6,7,0.10),
 circle(-6,7,0.12)
 )}}} 
 
 Graph of {{{y=(-11/6)x-4}}} through the points *[Tex \LARGE \left(0,-4\right)] and *[Tex \LARGE \left(-6,7\right)]