Question 119134
First lets find the slope through the points ({{{2}}},{{{-3}}}) and ({{{-1}}},{{{-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 ({{{2}}},{{{-3}}}) and  *[Tex \Large \left(x_{2},y_{2}\right)] is the second point ({{{-1}}},{{{-2}}}))


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


{{{m= 1/-3}}} Subtract the terms in the numerator {{{-2--3}}} to get {{{1}}}.  Subtract the terms in the denominator {{{-1-2}}} 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--3=(-1/3)(x-2)}}} Plug in {{{m=-1/3}}}, {{{x[1]=2}}}, and {{{y[1]=-3}}} (these values are given)



{{{y+3=(-1/3)(x-2)}}} Rewrite {{{y--3}}} as {{{y+3}}}



{{{y+3=(-1/3)x+(-1/3)(-2)}}} Distribute {{{-1/3}}}


{{{y+3=(-1/3)x+2/3}}} Multiply {{{-1/3}}} and {{{-2}}} to get {{{2/3}}}


{{{y=(-1/3)x+2/3-3}}} Subtract {{{3}}} from  both sides to isolate y


{{{y=(-1/3)x-7/3}}} Combine like terms {{{2/3}}} and {{{-3}}} to get {{{-7/3}}} (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 ({{{2}}},{{{-3}}}) and ({{{-1}}},{{{-2}}})  is:{{{y=(-1/3)x-7/3}}}


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=-7/3}}}


Notice if we graph the equation {{{y=(-1/3)x-7/3}}} and plot the points ({{{2}}},{{{-3}}}) and ({{{-1}}},{{{-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, -8.5, 9.5, -11.5, 6.5,
graph(500, 500, -8.5, 9.5, -11.5, 6.5,(-1/3)x+-7/3),
circle(2,-3,0.12),
circle(2,-3,0.12+0.03),
circle(-1,-2,0.12),
circle(-1,-2,0.12+0.03)
) }}} Graph of {{{y=(-1/3)x-7/3}}} through the points ({{{2}}},{{{-3}}}) and ({{{-1}}},{{{-2}}})


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