Question 142430
First lets find the slope through the points ({{{2}}},{{{5}}}) and ({{{7}}},{{{-3}}})


{{{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}}},{{{5}}}) and  *[Tex \Large \left(x_{2},y_{2}\right)] is the second point ({{{7}}},{{{-3}}}))


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


{{{m= -8/5}}} Subtract the terms in the numerator {{{-3-5}}} to get {{{-8}}}.  Subtract the terms in the denominator {{{7-2}}} to get {{{5}}}

  

So the slope is

{{{m=-8/5}}}


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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=(-8/5)(x-2)}}} Plug in {{{m=-8/5}}}, {{{x[1]=2}}}, and {{{y[1]=5}}} (these values are given)



{{{y-5=(-8/5)x+(-8/5)(-2)}}} Distribute {{{-8/5}}}


{{{y-5=(-8/5)x+16/5}}} Multiply {{{-8/5}}} and {{{-2}}} to get {{{16/5}}}


{{{y=(-8/5)x+16/5+5}}} Add {{{5}}} to  both sides to isolate y


{{{y=(-8/5)x+41/5}}} Combine like terms {{{16/5}}} and {{{5}}} to get {{{41/5}}} (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}}},{{{5}}}) and ({{{7}}},{{{-3}}})  is:{{{y=(-8/5)x+41/5}}}


The equation is now in {{{y=mx+b}}} form (which is slope-intercept form) where the slope is {{{m=-8/5}}} and the y-intercept is {{{b=41/5}}}


Notice if we graph the equation {{{y=(-8/5)x+41/5}}} and plot the points ({{{2}}},{{{5}}}) and ({{{7}}},{{{-3}}}),  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, -4.5, 13.5, -8, 10,
graph(500, 500, -4.5, 13.5, -8, 10,(-8/5)x+41/5),
circle(2,5,0.12),
circle(2,5,0.12+0.03),
circle(7,-3,0.12),
circle(7,-3,0.12+0.03)
) }}} Graph of {{{y=(-8/5)x+41/5}}} through the points ({{{2}}},{{{5}}}) and ({{{7}}},{{{-3}}})


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