SOLUTION: i know that slope intercept form is y=mx+b. so i find the slope of the second line then the negative reciprocal of that is the slope of the first line. after that, how do i find th

Algebra ->  Linear-equations -> SOLUTION: i know that slope intercept form is y=mx+b. so i find the slope of the second line then the negative reciprocal of that is the slope of the first line. after that, how do i find th      Log On


   

Question 93248: i know that slope intercept form is y=mx+b. so i find the slope of the second line then the negative reciprocal of that is the slope of the first line. after that, how do i find the y intercept to replace the b. i need some help please. this is the original question: Find the slope-intercept form of the equation of the line described. A line passing through the point (5,8) and perpendicular to the line passing though the points (-8,7) and (-2,0).
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First find the equation of the line passing through (-8,7) and (-2,0)


We can find the slope through the points (-8,7) and (-2,0) using this formula:

m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29

m=%280-7%29%2F%28-2--8%29 Plug in y%5B2%5D=0,y%5B1%5D=7,x%5B2%5D=-2,x%5B1%5D=-8 (these are the coordinates of given points)

m=+-7%2F6 Subtract the terms in the numerator 0-7 to get -7. Subtract the terms in the denominator -2--8 to get 6

So the slope is
m=-7%2F6

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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 is one of the given points

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

y-7=%28-7%2F6%29%28x--8%29 Plug in m=-7%2F6, x%5B1%5D=-8, and y%5B1%5D=7 (these values are given)


y-7=%28-7%2F6%29%28x%2B8%29 Rewrite x--8 as x%2B8


y-7=%28-7%2F6%29x%2B%28-7%2F6%29%288%29 Distribute -7%2F6

y-7=%28-7%2F6%29x-28%2F3 Multiply -7%2F6 and 8 to get -56%2F6. Now reduce -56%2F6 to get -28%2F3

y=%28-7%2F6%29x-28%2F3%2B7 Add 7 to both sides to isolate y

y=%28-7%2F6%29x-7%2F3 Combine like terms -28%2F3 and 7 to get -7%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 (-8,7) and (-2,0) is:y=%28-7%2F6%29x-7%2F3

The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=-7%2F6 and the y-intercept is b=-7%2F3

Notice if we graph the equation y=%28-7%2F6%29x-7%2F3 and plot the points (-8,7) and (-2,0), we get this: (note: if you need help with graphing, check out this solver)

Graph of y=%28-7%2F6%29x-7%2F3 through the points (-8,7) and (-2,0)


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Now lets find the equation of the perpendicular line through (5,8)


Solved by pluggable solver: Finding the Equation of a Line Parallel or Perpendicular to a Given Line


Remember, any two perpendicular lines are negative reciprocals of each other. So if you're given the slope of -7%2F6, you can find the perpendicular slope by this formula:

m%5Bp%5D=-1%2Fm where m%5Bp%5D is the perpendicular slope


m%5Bp%5D=-1%2F%28-7%2F6%29 So plug in the given slope to find the perpendicular slope



m%5Bp%5D=%28-1%2F1%29%286%2F-7%29 When you divide fractions, you multiply the first fraction (which is really 1%2F1) by the reciprocal of the second



m%5Bp%5D=6%2F7 Multiply the fractions.


So the perpendicular slope is 6%2F7



So now we know the slope of the unknown line is 6%2F7 (its the negative reciprocal of -7%2F6 from the line y=%28-7%2F6%29%2Ax-7%2F3). Also since the unknown line goes through (5,8), we can find the equation by plugging in this info into the point-slope formula

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 the given point



y-8=%286%2F7%29%2A%28x-5%29 Plug in m=6%2F7, x%5B1%5D=5, and y%5B1%5D=8



y-8=%286%2F7%29%2Ax-%286%2F7%29%285%29 Distribute 6%2F7



y-8=%286%2F7%29%2Ax-30%2F7 Multiply



y=%286%2F7%29%2Ax-30%2F7%2B8Add 8 to both sides to isolate y

y=%286%2F7%29%2Ax-30%2F7%2B56%2F7 Make into equivalent fractions with equal denominators



y=%286%2F7%29%2Ax%2B26%2F7 Combine the fractions



y=%286%2F7%29%2Ax%2B26%2F7 Reduce any fractions

So the equation of the line that is perpendicular to y=%28-7%2F6%29%2Ax-7%2F3 and goes through (5,8) is y=%286%2F7%29%2Ax%2B26%2F7


So here are the graphs of the equations y=%28-7%2F6%29%2Ax-7%2F3 and y=%286%2F7%29%2Ax%2B26%2F7




graph of the given equation y=%28-7%2F6%29%2Ax-7%2F3 (red) and graph of the line y=%286%2F7%29%2Ax%2B26%2F7(green) that is perpendicular to the given graph and goes through (5,8)