SOLUTION: Write and equation of the line containing (5,-2) and (8,4) a. 2x-y=12 b.x-2y=9 c.2x+y=20

Algebra ->  Coordinate Systems and Linear Equations  -> Lessons -> SOLUTION: Write and equation of the line containing (5,-2) and (8,4) a. 2x-y=12 b.x-2y=9 c.2x+y=20      Log On


   

Question 107155This question is from textbook Algebra and Trigonometry Structure And Method
: Write and equation of the line containing (5,-2) and (8,4)
a. 2x-y=12
b.x-2y=9
c.2x+y=20 This question is from textbook Algebra and Trigonometry Structure And Method

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 (8,4)


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


m=%284--2%29%2F%288-5%29 Plug in y%5B2%5D=4,y%5B1%5D=-2,x%5B2%5D=8,x%5B1%5D=5 (these are the coordinates of given points)


m=+6%2F3 Subtract the terms in the numerator 4--2 to get 6. Subtract the terms in the denominator 8-5 to get 3




m=2 Reduce



So the slope is

m=2





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


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



y%2B2=%282%29%28x-5%29 Rewrite y--2 as y%2B2



y%2B2=2x%2B%282%29%28-5%29 Distribute 2


y%2B2=2x-10 Multiply 2 and -5 to get -10%2F1. Now reduce -10%2F1 to get -10

y=2x-10-2 Subtract 2 from both sides to isolate y


y=2x-12 Combine like terms -10 and -2 to get -12

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

Answer:



So the equation of the line which goes through the points (5,-2) and (8,4) is:y=2x-12


The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=2 and the y-intercept is b=-12


Notice if we graph the equation y=2x-12 and plot the points (5,-2) and (8,4), we get this: (note: if you need help with graphing, check out this solver)


Graph of y=2x-12 through the points (5,-2) and (8,4)


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






Now lets convert the slope intercept equation into standard form


Solved by pluggable solver: Converting Linear Equations in Standard form to Slope-Intercept Form (and vice versa)
Convert from slope-intercept form (y = mx+b) to standard form (Ax+By = C)


y+=+2x-12 Start with the given equation


1y-2x+=+2x-12-2x Subtract 2x from both sides


-2x%2B1y+=+-12 Simplify


-1%2A%28-2x%2B1y%29+=+-1%2A%28-12%29 Multiply both sides by -1 to make the A coefficient positive (note: this step may be optional; it will depend on your teacher and/or textbook)


2x-1y+=+12 Distribute and simplify


The original equation y+=+2x-12 (slope-intercept form) is equivalent to 2x-1y+=+12 (standard form where A > 0)


The equation 2x-1y+=+12 is in the form Ax%2BBy+=+C where A+=+2, B+=+-1 and C+=+12






-1%28-2x%2By%29=-1%2A-12 Now multiply both sides by -1 to make A positive


2x-y=12 Distribute


So the equation is 2x-y=12 which means the answer is A)