Question 115649
If you only want to graph the line, simply plot the two points (2,1) and (4,7) and draw a straight line through them. 




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However, if you want to find the equation of the line, then...


First lets find the slope through the points ({{{2}}},{{{1}}}) and ({{{4}}},{{{7}}})


{{{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}}},{{{1}}}) and  *[Tex \Large \left(x_{2},y_{2}\right)] is the second point ({{{4}}},{{{7}}}))


{{{m=(7-1)/(4-2)}}} Plug in {{{y[2]=7}}},{{{y[1]=1}}},{{{x[2]=4}}},{{{x[1]=2}}}  (these are the coordinates of given points)


{{{m= 6/2}}} Subtract the terms in the numerator {{{7-1}}} to get {{{6}}}.  Subtract the terms in the denominator {{{4-2}}} to get {{{2}}}

  


{{{m=3}}} Reduce

  

So the slope is

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



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


{{{y-1=3x-6}}} Multiply {{{3}}} and {{{-2}}} to get {{{-6}}}


{{{y=3x-6+1}}} Add {{{1}}} to  both sides to isolate y


{{{y=3x-5}}} Combine like terms {{{-6}}} and {{{1}}} to get {{{-5}}} 

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Answer:



So the equation of the line which goes through the points ({{{2}}},{{{1}}}) and ({{{4}}},{{{7}}})  is:{{{y=3x-5}}}


The equation is now in {{{y=mx+b}}} form (which is slope-intercept form) where the slope is {{{m=3}}} and the y-intercept is {{{b=-5}}}


Notice if we graph the equation {{{y=3x-5}}} and plot the points ({{{2}}},{{{1}}}) and ({{{4}}},{{{7}}}),  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, -5, 13,
graph(500, 500, -6, 12, -5, 13,(3)x+-5),
circle(2,1,0.12),
circle(2,1,0.12+0.03),
circle(4,7,0.12),
circle(4,7,0.12+0.03)
) }}} Graph of {{{y=3x-5}}} through the points ({{{2}}},{{{1}}}) and ({{{4}}},{{{7}}})


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