Question 118099
First lets find the slope through the points ({{{4}}},{{{3}}}) and ({{{2}}},{{{-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 ({{{4}}},{{{3}}}) and  *[Tex \Large \left(x_{2},y_{2}\right)] is the second point ({{{2}}},{{{-2}}}))


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


{{{m= -5/-2}}} Subtract the terms in the numerator {{{-2-3}}} to get {{{-5}}}.  Subtract the terms in the denominator {{{2-4}}} to get {{{-2}}}

  


{{{m=5/2}}} Reduce

  

So the slope is

{{{m=5/2}}}


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



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=(5/2)(x-4)}}} Plug in {{{m=5/2}}}, {{{x[1]=4}}}, and {{{y[1]=3}}} (these values are given)



{{{y-3=(5/2)x+(5/2)(-4)}}} Distribute {{{5/2}}}


{{{y-3=(5/2)x-10}}} Multiply {{{5/2}}} and {{{-4}}} to get {{{-20/2}}}. Now reduce {{{-20/2}}} to get {{{-10}}}


{{{y=(5/2)x-10+3}}} Add {{{3}}} to  both sides to isolate y


{{{y=(5/2)x-7}}} Combine like terms {{{-10}}} and {{{3}}} to get {{{-7}}} 

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

Answer:



So the equation of the line which goes through the points ({{{4}}},{{{3}}}) and ({{{2}}},{{{-2}}})  is:{{{y=(5/2)x-7}}}


The equation is now in {{{y=mx+b}}} form (which is slope-intercept form) where the slope is {{{m=5/2}}} and the y-intercept is {{{b=-7}}}


Notice if we graph the equation {{{y=(5/2)x-7}}} and plot the points ({{{4}}},{{{3}}}) and ({{{2}}},{{{-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, -6, 12, -8.5, 9.5,
graph(500, 500, -6, 12, -8.5, 9.5,(5/2)x+-7),
circle(4,3,0.12),
circle(4,3,0.12+0.03),
circle(2,-2,0.12),
circle(2,-2,0.12+0.03)
) }}} Graph of {{{y=(5/2)x-7}}} through the points ({{{4}}},{{{3}}}) and ({{{2}}},{{{-2}}})


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