SOLUTION: Find an equation of the line passing through each pair of points. Write the equation in the form Ax+By=C (5,-2)and(3,7)

Algebra ->  Linear-equations -> SOLUTION: Find an equation of the line passing through each pair of points. Write the equation in the form Ax+By=C (5,-2)and(3,7)      Log On


   

Question 79782: Find an equation of the line passing through each pair of points. Write the equation in the form Ax+By=C
(5,-2)and(3,7)
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 (5,-2) and (3,7)


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 (5,-2) and (x%5B2%5D,y%5B2%5D) is the second point (3,7))


m=%287--2%29%2F%283-5%29 Plug in y%5B2%5D=7,y%5B1%5D=-2,x%5B2%5D=3,x%5B1%5D=5 (these are the coordinates of given points)


m=+9%2F-2 Subtract the terms in the numerator 7--2 to get 9. Subtract the terms in the denominator 3-5 to get -2




m=-9%2F2 Reduce



So the slope is

m=-9%2F2





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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 (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--2=%28-9%2F2%29%28x-5%29 Plug in m=-9%2F2, x%5B1%5D=5, and y%5B1%5D=-2 (these values are given)



y%2B2=%28-9%2F2%29%28x-5%29 Rewrite y--2 as y%2B2



y%2B2=%28-9%2F2%29x%2B%28-9%2F2%29%28-5%29 Distribute -9%2F2


y%2B2=%28-9%2F2%29x%2B45%2F2 Multiply -9%2F2 and -5 to get 45%2F2

y=%28-9%2F2%29x%2B45%2F2-2 Subtract 2 from both sides to isolate y


y=%28-9%2F2%29x%2B41%2F2 Combine like terms 45%2F2 and -2 to get 41%2F2 (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 (5,-2) and (3,7) is:y=%28-9%2F2%29x%2B41%2F2


The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=-9%2F2 and the y-intercept is b=41%2F2


Notice if we graph the equation y=%28-9%2F2%29x%2B41%2F2 and plot the points (5,-2) and (3,7), we get this: (note: if you need help with graphing, check out this solver)


Graph of y=%28-9%2F2%29x%2B41%2F2 through the points (5,-2) and (3,7)


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




Now to get it into standard form, add 4.5x to both sides

y%2B4.5x=cross%28-4.5x%2B4.5x%29%2B20.5

y%2B4.5x=20.5

Now multiply both sides by 2
2%28y%2B4.5x%29=2%2820.5%29

2y%2B9x=41
So it is now in standard form if we arrange x and y

9x%2B2y=41

where A=9, B=2, and C=41