SOLUTION: Determine the algebraic rule for the line that: contains (1/2, 3) and (2, -1)

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Question 124960This question is from textbook Intermediate Algebra Review
: Determine the algebraic rule for the line that:
contains (1/2, 3) and (2, -1) This question is from textbook Intermediate Algebra Review

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First lets find the slope through the points (1%2F2,3) and (2,-1)

m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula (note: is the first point (1%2F2,3) and is the second point (2,-1))

m=%28-1-3%29%2F%282-1%2F2%29 Plug in y%5B2%5D=-1,y%5B1%5D=3,x%5B2%5D=2,x%5B1%5D=1%2F2 (these are the coordinates of given points)

m=+%28-4+%2F+1%29%2F%283+%2F+2%29 Subtract (note: if you need help with subtracting or dividing fractions, check out this solver)



m=-8+%2F+3 Divide the fractions

So the slope is
m=-8+%2F+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%5B1%5D=m%28x-x%5B1%5D%29 where m is the slope, and is one of the given points

So lets use the Point-Slope Formula to find the equation of the line

y-3=%28-8+%2F+3%29%28x-1%2F2%29 Plug in m=-8+%2F+3, x%5B1%5D=1%2F2, and y%5B1%5D=3 (these values are given)


y-3=%28-8%2F3%29x%2B%28-8%2F3%29%28-1%2F2%29 Distribute -8+%2F+3

y-3=%28-8%2F3%29x%2B4%2F3 Multiply -8+%2F+3 and -1%2F2 to get 8%2F6. Now reduce 8%2F6 to get 4%2F3

y=%28-8%2F3%29x%2B4%2F3%2B3 Add 3 to both sides to isolate y

y=%28-8%2F3%29x%2B13%2F3 Combine like terms 4%2F3 and 3 to get 13%2F3 (note: if you need help with combining fractions, check out this solver)


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Answer:


So the equation of the line which goes through the points (1%2F2,3) and (2,-1) is:y=%28-8%2F3%29x%2B13%2F3

The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=-8+%2F+3 and the y-intercept is b=13%2F3

Notice if we graph the equation y=%28-8%2F3%29x%2B13%2F3 and plot the points (1%2F2,3) and (2,-1), we get this: (note: if you need help with graphing, check out this solver)

Graph of y=%28-8%2F3%29x%2B13%2F3 through the points (1%2F2,3) and (2,-1)

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