Question 113890
First lets find the slope through the points ({{{6}}},{{{-5}}}) and ({{{-3}}},{{{4}}})


{{{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 ({{{6}}},{{{-5}}}) and  *[Tex \Large \left(x_{2},y_{2}\right)] is the second point ({{{-3}}},{{{4}}}))


{{{m=(4--5)/(-3-6)}}} Plug in {{{y[2]=4}}},{{{y[1]=-5}}},{{{x[2]=-3}}},{{{x[1]=6}}}  (these are the coordinates of given points)


{{{m= 9/-9}}} Subtract the terms in the numerator {{{4--5}}} to get {{{9}}}.  Subtract the terms in the denominator {{{-3-6}}} to get {{{-9}}}

  


{{{m=-1}}} Reduce

  

So the slope is

{{{m=-1}}}


------------------------------------------------



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=(-1)(x-6)}}} Plug in {{{m=-1}}}, {{{x[1]=6}}}, and {{{y[1]=-5}}} (these values are given)



{{{y+5=(-1)(x-6)}}} Rewrite {{{y--5}}} as {{{y+5}}}



{{{y+5=-x+(-1)(-6)}}} Distribute {{{-1}}}


{{{y+5=-x+6}}} Multiply {{{-1}}} and {{{-6}}} to get {{{6}}}


{{{y=-x+6-5}}} Subtract {{{5}}} from  both sides to isolate y


{{{y=-x+1}}} Combine like terms {{{6}}} and {{{-5}}} to get {{{1}}} 

------------------------------------------------------------------------------------------------------------

Answer:



So the equation of the line which goes through the points ({{{6}}},{{{-5}}}) and ({{{-3}}},{{{4}}})  is:{{{y=-x+1}}}


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


Notice if we graph the equation {{{y=-x+1}}} and plot the points ({{{6}}},{{{-5}}}) and ({{{-3}}},{{{4}}}),  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, -7.5, 10.5, -9.5, 8.5,
graph(500, 500, -7.5, 10.5, -9.5, 8.5,(-1)x+1),
circle(6,-5,0.12),
circle(6,-5,0.12+0.03),
circle(-3,4,0.12),
circle(-3,4,0.12+0.03)
) }}} Graph of {{{y=-x+1}}} through the points ({{{6}}},{{{-5}}}) and ({{{-3}}},{{{4}}})


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