SOLUTION: The equation of the line through (8,6) and (2,-4) is A. 5x - 3y = 22 B. Y = 3/5x = 8/5 C. 3x = 4y = 48 D. -4x + 2y = -16

Algebra ->  Algebra  -> College  -> Linear Algebra -> SOLUTION: The equation of the line through (8,6) and (2,-4) is A. 5x - 3y = 22 B. Y = 3/5x = 8/5 C. 3x = 4y = 48 D. -4x + 2y = -16       Log On


   

Question 87170: The equation of the line through (8,6) and (2,-4) is


A. 5x - 3y = 22
B. Y = 3/5x = 8/5
C. 3x = 4y = 48
D. -4x + 2y = -16



Answer by jim_thompson5910(21667) 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 (8,6) and (2,-4)


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 (8,6) and (x%5B2%5D,y%5B2%5D) is the second point (2,-4))


m=%28-4-6%29%2F%282-8%29 Plug in y%5B2%5D=-4,y%5B1%5D=6,x%5B2%5D=2,x%5B1%5D=8 (these are the coordinates of given points)


m=+-10%2F-6 Subtract the terms in the numerator -4-6 to get -10. Subtract the terms in the denominator 2-8 to get -6




m=5%2F3 Reduce



So the slope is

m=5%2F3


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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-6=%285%2F3%29%28x-8%29 Plug in m=5%2F3, x%5B1%5D=8, and y%5B1%5D=6 (these values are given)



y-6=%285%2F3%29x%2B%285%2F3%29%28-8%29 Distribute 5%2F3


y-6=%285%2F3%29x-40%2F3 Multiply 5%2F3 and -8 to get -40%2F3

y=%285%2F3%29x-40%2F3%2B6 Add 6 to both sides to isolate y


y=%285%2F3%29x-22%2F3 Combine like terms -40%2F3 and 6 to get -22%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 (8,6) and (2,-4) is:y=%285%2F3%29x-22%2F3


The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=5%2F3 and the y-intercept is b=-22%2F3


Notice if we graph the equation y=%285%2F3%29x-22%2F3 and plot the points (8,6) and (2,-4), we get this: (note: if you need help with graphing, check out this solver)


drawing%28500%2C+500%2C+-4%2C+14%2C+-8%2C+10%2C%0D%0A++graph%28500%2C+500%2C+-4%2C+14%2C+-8%2C+10%2C%285%2F3%29x%2B-22%2F3%29%2C%0D%0A++circle%288%2C6%2C0.12%29%2C%0D%0A++circle%288%2C6%2C0.12%2B0.03%29%2C%0D%0A++circle%282%2C-4%2C0.12%29%2C%0D%0A++circle%282%2C-4%2C0.12%2B0.03%29%0D%0A++%29+ Graph of y=%285%2F3%29x-22%2F3 through the points (8,6) and (2,-4)


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





Since none of these answers match the equation in slope-intercept form, lets convert it to standard form:

Solved by pluggable solver: Converting Linear Equations in Standard form to Slope-Intercept Form (and vice versa)


y=%285%2F3%29%2Ax-22%2F3Start with the given equation



y-%285%2F3%29%2Ax=-22%2F3Subtract %285%2F3%29%2Ax from both sides

-%285%2F3%29%2Ax%2By=-22%2F3 Rearrange

3%28-%285%2F3%29%2Ax%2By%29=3%28-22%2F3%29 Multiply both sides by 3



3%28-5%2F3%29x%2B3%2Ay=3%28-22%2F3%29 Distribute



-%2815%2F3%29%2Ax%2B3%2Ay=-66%2F3 Multiply

-5%2Ax%2B3%2Ay=-22 Reduce

Now the equation is in standard form Ax%2BBy=C where A=-5, B=3, and C=-22




-1%28-5x%2B3y%29=-1%2A-22 Multiply both sides by -1 to make C a positive number

5x-3y=22

So the answer is A