Question 217337
Find the slope-intercept form of the equation of the line that passes through the given points. 


(2, 2)
(6, -2/3) 


Step 1.  We need to find an equation in slope intercept form given as y=mx+b where m is the slope and b is the y-intercept at point (0,b).


Step 2.  The slope of the line m is given as


{{{ m=(y2-y1)/(x2-x1)}}}


where for our example is x1=2, y1=2, x2=6 and y2=-2/3 (think of {{{slope=rise/run}}}).  You can choose the points the other way around but be consistent with the x and y coordinates.  You will get the same result.


Step 3.  Substituting the above values in the slope equation gives


{{{m=(-2/3-2)/(6-2)}}}


{{{m=(-8/3)/4}}}


{{{m=-2/3}}}


Step 4.  The slope is calculated as {{{-2/3}}} or {{{m=-2/3}}}.


Step 5.  Now use the slope equation of step 1 and choose one of the given points.  I'll choose point (2,2).   Letting y=y2 and x=x2 and substituting m=60 in the slope equation given as,


{{{ m=(y2-y1)/(x2-x1)}}}



{{{ -2/3=(y-2)/(x-2)}}}


{{{ -2/3=(y-2)/(x-2)}}}


Step 6.  Multiply both sides of equation by 3(x-2) to get rid of denomination found on the right side of the equation



{{{ -2(x-2)/3=3(x-2)(y-2)/(x-2)}}}



{{{ -2x+4=3y-6}}}



Step 7.  Now simplify and put the above equation into slope-intercept form.


{{{-2x+4=3y-6}}}


Add 6 to both sides of the equation


{{{-2x+4+6=3y-6+6}}}


{{{-2x+10=3y}}}


Divide by 3 to both sides of the equation


{{{-2x/3+10/3=3y/3}}}


{{{-2x/3+10/3=y}}}   



ANSWER in slope-intercept form is {{{y=-2x/3+10/3}}} where slope m=-2/3 and y-intercept=10/3


Step 8.  See if the other point (6,-2/3) or x=5 and y=-2/3 satisfies this equation


{{{y=-2x/3+10/3}}}


{{{-2/3=-2/3*6+10/3=-2/3}}}  So the other point satisfies this equation and lies on the line.


In other words, you can use the other point to check your work.


Note: above equation can be also be transform into standard form as


{{{2x+3y=10}}}


See graph below to check the above steps.  Note the slope and y-intercept as well as the x-intercept.


{{{graph(400,400,-10,10,-10,10,-2x/3+10/3)}}}


I hope the above steps were helpful. 

 
And good luck in your studies!


For free Step-By-Step Videos on Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra or for Trigonometry visit http://www.FreedomUniversity.TV/courses/Trigonometry.


Respectfully,
Dr J