SOLUTION: How do you write the equation for a line passing through each pair of points? example: (3,5) (-3,1)

Algebra ->  Linear-equations -> SOLUTION: How do you write the equation for a line passing through each pair of points? example: (3,5) (-3,1)      Log On


   

Question 82667: How do you write the equation for a line passing through each pair of points?
example:
(3,5) (-3,1)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Finding the Equation of a Line
First lets find the slope through the points (3,5) and (-3,1)


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


m=%281-5%29%2F%28-3-3%29 Plug in y%5B2%5D=1,y%5B1%5D=5,x%5B2%5D=-3,x%5B1%5D=3 (these are the coordinates of given points)


m=+-4%2F-6 Subtract the terms in the numerator 1-5 to get -4. Subtract the terms in the denominator -3-3 to get -6




m=2%2F3 Reduce



So the slope is

m=2%2F3





------------------------------------------------


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 (x%5B1%5D,y%5B1%5D) is one of the given points


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


y-5=%282%2F3%29%28x-3%29 Plug in m=2%2F3, x%5B1%5D=3, and y%5B1%5D=5 (these values are given)



y-5=%282%2F3%29x%2B%282%2F3%29%28-3%29 Distribute 2%2F3


y-5=%282%2F3%29x-2 Multiply 2%2F3 and -3 to get -6%2F3. Now reduce -6%2F3 to get -2

y=%282%2F3%29x-2%2B5 Add 5 to both sides to isolate y


y=%282%2F3%29x%2B3 Combine like terms -2 and 5 to get 3

------------------------------------------------------------------------------------------------------------

Answer:



So the equation of the line which goes through the points (3,5) and (-3,1) is:y=%282%2F3%29x%2B3


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


Notice if we graph the equation y=%282%2F3%29x%2B3 and plot the points (3,5) and (-3,1), we get this: (note: if you need help with graphing, check out this solver)


Graph of y=%282%2F3%29x%2B3 through the points (3,5) and (-3,1)


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