SOLUTION: For the line represented by 3x-2y=24, give an equation of the line perpendicular to it which passes through the point (7,-5).

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Question 487246: For the line represented by 3x-2y=24, give an equation of the line perpendicular to it which passes through the point (7,-5).
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
3x-2y=24,........->... 3x-24=2y...->... y=%283%2F2%29x-12

passes through the point (7,-5)

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 3%2F2, 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%283%2F2%29 So plug in the given slope to find the perpendicular slope



m%5Bp%5D=%28-1%2F1%29%282%2F3%29 When you divide fractions, you multiply the first fraction (which is really 1%2F1) by the reciprocal of the second



m%5Bp%5D=-2%2F3 Multiply the fractions.


So the perpendicular slope is -2%2F3



So now we know the slope of the unknown line is -2%2F3 (its the negative reciprocal of 3%2F2 from the line y=%283%2F2%29%2Ax-12). Also since the unknown line goes through (7,-5), 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%2B5=%28-2%2F3%29%2A%28x-7%29 Plug in m=-2%2F3, x%5B1%5D=7, and y%5B1%5D=-5



y%2B5=%28-2%2F3%29%2Ax%2B%282%2F3%29%287%29 Distribute -2%2F3



y%2B5=%28-2%2F3%29%2Ax%2B14%2F3 Multiply



y=%28-2%2F3%29%2Ax%2B14%2F3-5Subtract -5 from both sides to isolate y

y=%28-2%2F3%29%2Ax%2B14%2F3-15%2F3 Make into equivalent fractions with equal denominators



y=%28-2%2F3%29%2Ax-1%2F3 Combine the fractions



y=%28-2%2F3%29%2Ax-1%2F3 Reduce any fractions

So the equation of the line that is perpendicular to y=%283%2F2%29%2Ax-12 and goes through (7,-5) is y=%28-2%2F3%29%2Ax-1%2F3


So here are the graphs of the equations y=%283%2F2%29%2Ax-12 and y=%28-2%2F3%29%2Ax-1%2F3




graph of the given equation y=%283%2F2%29%2Ax-12 (red) and graph of the line y=%28-2%2F3%29%2Ax-1%2F3(green) that is perpendicular to the given graph and goes through (7,-5)