Question 119020
First lets find the slope through the points ({{{1}}},{{{-5}}}) and ({{{-3}}},{{{7}}})


{{{m=(y[2]-y[1])/(x[2]-x[1])}}} Start with the slope formula (note: *[Tex \Large \left(x_{1},y_{1}\right)] is the first point ({{{1}}},{{{-5}}}) and  *[Tex \Large \left(x_{2},y_{2}\right)] is the second point ({{{-3}}},{{{7}}}))


{{{m=(7--5)/(-3-1)}}} Plug in {{{y[2]=7}}},{{{y[1]=-5}}},{{{x[2]=-3}}},{{{x[1]=1}}}  (these are the coordinates of given points)


{{{m= 12/-4}}} Subtract the terms in the numerator {{{7--5}}} to get {{{12}}}.  Subtract the terms in the denominator {{{-3-1}}} to get {{{-4}}}

  


{{{m=-3}}} Reduce

  

So the slope is

{{{m=-3}}}


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Now let's use the point-slope formula to find the equation of the line:




------Point-Slope Formula------
{{{y-y[1]=m(x-x[1])}}} where {{{m}}} is the slope, and *[Tex \Large \left(\textrm{x_{1},y_{1}}\right)] is one of the given points


So lets use the Point-Slope Formula to find the equation of the line


{{{y--5=(-3)(x-1)}}} Plug in {{{m=-3}}}, {{{x[1]=1}}}, and {{{y[1]=-5}}} (these values are given)



{{{y+5=(-3)(x-1)}}} Rewrite {{{y--5}}} as {{{y+5}}}



{{{y+5=-3x+(-3)(-1)}}} Distribute {{{-3}}}


{{{y+5=-3x+3}}} Multiply {{{-3}}} and {{{-1}}} to get {{{3}}}


{{{y=-3x+3-5}}} Subtract {{{5}}} from  both sides to isolate y


{{{y=-3x-2}}} Combine like terms {{{3}}} and {{{-5}}} to get {{{-2}}} 

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Answer:



So the equation of the line which goes through the points ({{{1}}},{{{-5}}}) and ({{{-3}}},{{{7}}})  is:{{{y=-3x-2}}}


The equation is now in {{{y=mx+b}}} form (which is slope-intercept form) where the slope is {{{m=-3}}} and the y-intercept is {{{b=-2}}}


Notice if we graph the equation {{{y=-3x-2}}} and plot the points ({{{1}}},{{{-5}}}) and ({{{-3}}},{{{7}}}),  we get this: (note: if you need help with graphing, check out this <a href=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver<a>)


{{{drawing(500, 500, -10, 8, -8, 10,
graph(500, 500, -10, 8, -8, 10,(-3)x+-2),
circle(1,-5,0.12),
circle(1,-5,0.12+0.03),
circle(-3,7,0.12),
circle(-3,7,0.12+0.03)
) }}} Graph of {{{y=-3x-2}}} through the points ({{{1}}},{{{-5}}}) and ({{{-3}}},{{{7}}})


Notice how the two points lie on the line. This graphically verifies our answer.