Question 124960
First lets find the slope through the points ({{{1/2}}},{{{3}}}) and ({{{2}}},{{{-1}}})


{{{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 ({{{1/2}}},{{{3}}}) and  *[Tex \Large \left(x_{2},y_{2}\right)] is the second point ({{{2}}},{{{-1}}}))


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


{{{m= (-4 / 1)/(3 / 2)}}} Subtract   (note: if you need help with subtracting or dividing fractions, check out this <a href=http://www.algebra.com/algebra/homework/NumericFractions/fractions-solver.solver>solver</a>)




{{{m=-8 / 3}}} Divide the fractions

  

So the slope is

{{{m=-8 / 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=(-8 / 3)(x-1/2)}}} Plug in {{{m=-8 / 3}}}, {{{x[1]=1/2}}}, and {{{y[1]=3}}} (these values are given)



{{{y-3=(-8/3)x+(-8/3)(-1/2)}}} Distribute {{{-8 / 3}}}


{{{y-3=(-8/3)x+4/3}}} Multiply {{{-8 / 3}}} and {{{-1/2}}} to get {{{8/6}}}. Now reduce {{{8/6}}} to get {{{4/3}}}


{{{y=(-8/3)x+4/3+3}}} Add {{{3}}} to  both sides to isolate y


{{{y=(-8/3)x+13/3}}} Combine like terms {{{4/3}}} and {{{3}}} to get {{{13/3}}} (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 ({{{1/2}}},{{{3}}}) and ({{{2}}},{{{-1}}})  is:{{{y=(-8/3)x+13/3}}}


The equation is now in {{{y=mx+b}}} form (which is slope-intercept form) where the slope is {{{m=-8 / 3}}} and the y-intercept is {{{b=13/3}}}


Notice if we graph the equation {{{y=(-8/3)x+13/3}}} and plot the points ({{{1/2}}},{{{3}}}) and ({{{2}}},{{{-1}}}),  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, -8, 10,
graph(500, 500, -7.5, 10.5, -8, 10,(-8 / 3)x+13/3),
circle(1/2,3,0.12),
circle(1/2,3,0.12+0.03),
circle(2,-1,0.12),
circle(2,-1,0.12+0.03)
) }}} Graph of {{{y=(-8/3)x+13/3}}} through the points ({{{1/2}}},{{{3}}}) and ({{{2}}},{{{-1}}})


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