Question 122018
First lets find the slope through the points ({{{5}}},{{{3}}}) and ({{{-7}}},{{{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 ({{{5}}},{{{3}}}) and  *[Tex \Large \left(x_{2},y_{2}\right)] is the second point ({{{-7}}},{{{7}}}))


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


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

  


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



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


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


{{{y=(-1/3)x+5/3+3}}} Add {{{3}}} to  both sides to isolate y


{{{y=(-1/3)x+14/3}}} Combine like terms {{{5/3}}} and {{{3}}} to get {{{14/3}}} (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 ({{{5}}},{{{3}}}) and ({{{-7}}},{{{7}}})  is:{{{y=(-1/3)x+14/3}}}


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=14/3}}}


Notice if we graph the equation {{{y=(-1/3)x+14/3}}} and plot the points ({{{5}}},{{{3}}}) and ({{{-7}}},{{{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, -4, 14,
graph(500, 500, -10, 8, -4, 14,(-1/3)x+14/3),
circle(5,3,0.12),
circle(5,3,0.12+0.03),
circle(-7,7,0.12),
circle(-7,7,0.12+0.03)
) }}} Graph of {{{y=(-1/3)x+14/3}}} through the points ({{{5}}},{{{3}}}) and ({{{-7}}},{{{7}}})


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