Question 132918
Do you want to find the equation of the line through these points?




First lets find the slope through the points ({{{-3}}},{{{-2}}}) and ({{{5}}},{{{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 ({{{-3}}},{{{-2}}}) and  *[Tex \Large \left(x_{2},y_{2}\right)] is the second point ({{{5}}},{{{4}}}))


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


{{{m= 6/8}}} Subtract the terms in the numerator {{{4--2}}} to get {{{6}}}.  Subtract the terms in the denominator {{{5--3}}} to get {{{8}}}

  


{{{m=3/4}}} Reduce

  

So the slope is

{{{m=3/4}}}


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



{{{y+2=(3/4)(x--3)}}} Rewrite {{{y--2}}} as {{{y+2}}}



{{{y+2=(3/4)(x+3)}}} Rewrite {{{x--3}}} as {{{x+3}}}



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


{{{y+2=(3/4)x+9/4}}} Multiply {{{3/4}}} and {{{3}}} to get {{{9/4}}}


{{{y=(3/4)x+9/4-2}}} Subtract {{{2}}} from  both sides to isolate y


{{{y=(3/4)x+1/4}}} Combine like terms {{{9/4}}} and {{{-2}}} to get {{{1/4}}} (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 ({{{-3}}},{{{-2}}}) and ({{{5}}},{{{4}}})  is:{{{y=(3/4)x+1/4}}}


The equation is now in {{{y=mx+b}}} form (which is slope-intercept form) where the slope is {{{m=3/4}}} and the y-intercept is {{{b=1/4}}}


Notice if we graph the equation {{{y=(3/4)x+1/4}}} and plot the points ({{{-3}}},{{{-2}}}) and ({{{5}}},{{{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, -8, 10, -8, 10,
graph(500, 500, -8, 10, -8, 10,(3/4)x+1/4),
circle(-3,-2,0.12),
circle(-3,-2,0.12+0.03),
circle(5,4,0.12),
circle(5,4,0.12+0.03)
) }}} Graph of {{{y=(3/4)x+1/4}}} through the points ({{{-3}}},{{{-2}}}) and ({{{5}}},{{{4}}})


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