Question 130002
First lets find the slope through the points ({{{-1}}},{{{1}}}) and ({{{-4}}},{{{2}}})


{{{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}}},{{{1}}}) and  *[Tex \Large \left(x_{2},y_{2}\right)] is the second point ({{{-4}}},{{{2}}}))


{{{m=(2-1)/(-4--1)}}} Plug in {{{y[2]=2}}},{{{y[1]=1}}},{{{x[2]=-4}}},{{{x[1]=-1}}}  (these are the coordinates of given points)


{{{m= 1/-3}}} Subtract the terms in the numerator {{{2-1}}} to get {{{1}}}.  Subtract the terms in the denominator {{{-4--1}}} to get {{{-3}}}

  


{{{m=-1/3}}} Reduce

  

So the slope is

{{{m=-1/3}}}


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



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



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


{{{y-1=(-1/3)x-1/3}}} Multiply {{{-1/3}}} and {{{1}}} to get {{{-1/3}}}


{{{y=(-1/3)x-1/3+1}}} Add {{{1}}} to  both sides to isolate y


{{{y=(-1/3)x+2/3}}} Combine like terms {{{-1/3}}} and {{{1}}} to get {{{2/3}}} (note: if you need help with combining fractions, check out this <a href=http://www.algebra.com/algebra/homework/NumericFractions/fractions-solver.solver>solver</a>)



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Answer:



So the equation of the line which goes through the points ({{{-1}}},{{{1}}}) and ({{{-4}}},{{{2}}})  is:{{{y=(-1/3)x+2/3}}}


The equation is now in {{{y=mx+b}}} form (which is slope-intercept form) where the slope is {{{m=-1/3}}} and the y-intercept is {{{b=2/3}}}


Notice if we graph the equation {{{y=(-1/3)x+2/3}}} and plot the points ({{{-1}}},{{{1}}}) and ({{{-4}}},{{{2}}}),  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, -11.5, 6.5, -7.5, 10.5,
graph(500, 500, -11.5, 6.5, -7.5, 10.5,(-1/3)x+2/3),
circle(-1,1,0.12),
circle(-1,1,0.12+0.03),
circle(-4,2,0.12),
circle(-4,2,0.12+0.03)
) }}} Graph of {{{y=(-1/3)x+2/3}}} through the points ({{{-1}}},{{{1}}}) and ({{{-4}}},{{{2}}})


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