SOLUTION: Find the equation of the line through the points (4,-1) and (2,-7). Write the equation in standard form with only integers. Is standard form ax+by=c? If so, how do I go about

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: Find the equation of the line through the points (4,-1) and (2,-7). Write the equation in standard form with only integers. Is standard form ax+by=c? If so, how do I go about       Log On


   

Question 106545: Find the equation of the line through the points (4,-1) and (2,-7). Write the equation in standard form with only integers.
Is standard form ax+by=c? If so, how do I go about puting this in standard form?
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 (4,-1) and (2,-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 (4,-1) and (x%5B2%5D,y%5B2%5D) is the second point (2,-7))


m=%28-7--1%29%2F%282-4%29 Plug in y%5B2%5D=-7,y%5B1%5D=-1,x%5B2%5D=2,x%5B1%5D=4 (these are the coordinates of given points)


m=+-6%2F-2 Subtract the terms in the numerator -7--1 to get -6. Subtract the terms in the denominator 2-4 to get -2




m=3 Reduce



So the slope is

m=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 (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--1=%283%29%28x-4%29 Plug in m=3, x%5B1%5D=4, and y%5B1%5D=-1 (these values are given)



y%2B1=%283%29%28x-4%29 Rewrite y--1 as y%2B1



y%2B1=3x%2B%283%29%28-4%29 Distribute 3


y%2B1=3x-12 Multiply 3 and -4 to get -12%2F1. Now reduce -12%2F1 to get -12

y=3x-12-1 Subtract 1 from both sides to isolate y


y=3x-13 Combine like terms -12 and -1 to get -13

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Answer:



So the equation of the line which goes through the points (4,-1) and (2,-7) is:y=3x-13


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


Notice if we graph the equation y=3x-13 and plot the points (4,-1) and (2,-7), we get this: (note: if you need help with graphing, check out this solver)


Graph of y=3x-13 through the points (4,-1) and (2,-7)


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





Now let's convert the 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+=+3x-13 Start with the given equation


1y-3x+=+3x-13-3x Subtract 3x from both sides


-3x%2B1y+=+-13 Simplify


-1%2A%28-3x%2B1y%29+=+-1%2A%28-13%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)


3x-1y+=+13 Distribute and simplify


The original equation y+=+3x-13 (slope-intercept form) is equivalent to 3x-1y+=+13 (standard form where A > 0)


The equation 3x-1y+=+13 is in the form Ax%2BBy+=+C where A+=+3, B+=+-1 and C+=+13