SOLUTION: Write and equation in point-slope form for a line perpendicular to y=2/3x+1 and goes through the point (8,-4).

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Question 117662: Write and equation in point-slope form for a line perpendicular to y=2/3x+1 and goes through the point (8,-4).
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 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 2%2F3, 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%282%2F3%29 So plug in the given slope to find the perpendicular slope



m%5Bp%5D=%28-1%2F1%29%283%2F2%29 When you divide fractions, you multiply the first fraction (which is really 1%2F1) by the reciprocal of the second



m%5Bp%5D=-3%2F2 Multiply the fractions.


So the perpendicular slope is -3%2F2



So now we know the slope of the unknown line is -3%2F2 (its the negative reciprocal of 2%2F3 from the line y=%282%2F3%29%2Ax%2B1). Also since the unknown line goes through (8,-4), 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%2B4=%28-3%2F2%29%2A%28x-8%29 Plug in m=-3%2F2, x%5B1%5D=8, and y%5B1%5D=-4



y%2B4=%28-3%2F2%29%2Ax%2B%283%2F2%29%288%29 Distribute -3%2F2



y%2B4=%28-3%2F2%29%2Ax%2B24%2F2 Multiply



y=%28-3%2F2%29%2Ax%2B24%2F2-4Subtract -4 from both sides to isolate y

y=%28-3%2F2%29%2Ax%2B24%2F2-8%2F2 Make into equivalent fractions with equal denominators



y=%28-3%2F2%29%2Ax%2B16%2F2 Combine the fractions



y=%28-3%2F2%29%2Ax%2B8 Reduce any fractions

So the equation of the line that is perpendicular to y=%282%2F3%29%2Ax%2B1 and goes through (8,-4) is y=%28-3%2F2%29%2Ax%2B8


So here are the graphs of the equations y=%282%2F3%29%2Ax%2B1 and y=%28-3%2F2%29%2Ax%2B8




graph of the given equation y=%282%2F3%29%2Ax%2B1 (red) and graph of the line y=%28-3%2F2%29%2Ax%2B8(green) that is perpendicular to the given graph and goes through (8,-4)