SOLUTION: line Z goes through the point (2, -4) and is perpendicular to the line through (-6,5) and (-3, -2). What is the slope of Z? I don't even know where to start.

Algebra ->  Linear-equations -> SOLUTION: line Z goes through the point (2, -4) and is perpendicular to the line through (-6,5) and (-3, -2). What is the slope of Z? I don't even know where to start.      Log On


   

Question 98508: line Z goes through the point (2, -4) and is perpendicular to the line through (-6,5) and (-3, -2). What is the slope of Z? I don't even know where to start.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Lets find the equation of the line through (-6,5) and (-3, -2)


First lets find the slope through the points (-6,5) and (-3,-2)

m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula (note: is the first point (-6,5) and is the second point (-3,-2))

m=%28-2-5%29%2F%28-3--6%29 Plug in y%5B2%5D=-2,y%5B1%5D=5,x%5B2%5D=-3,x%5B1%5D=-6 (these are the coordinates of given points)

m=+-7%2F3 Subtract the terms in the numerator -2-5 to get -7. Subtract the terms in the denominator -3--6 to get 3

So the slope is
m=-7%2F3

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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-5=%28-7%2F3%29%28x--6%29 Plug in m=-7%2F3, x%5B1%5D=-6, and y%5B1%5D=5 (these values are given)


y-5=%28-7%2F3%29%28x%2B6%29 Rewrite x--6 as x%2B6


y-5=%28-7%2F3%29x%2B%28-7%2F3%29%286%29 Distribute -7%2F3

y-5=%28-7%2F3%29x-14 Multiply -7%2F3 and 6 to get -42%2F3. Now reduce -42%2F3 to get -14

y=%28-7%2F3%29x-14%2B5 Add 5 to both sides to isolate y

y=%28-7%2F3%29x-9 Combine like terms -14 and 5 to get -9
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Answer:


So the equation of the line which goes through the points (-6,5) and (-3,-2) is:y=%28-7%2F3%29x-9



Now let's find the equation of the line perpendicular to y=%28-7%2F3%29x-9 and goes through (2,-4)


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%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%28-7%2F3%29 So plug in the given slope to find the perpendicular slope



m%5Bp%5D=%28-1%2F1%29%283%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=3%2F7 Multiply the fractions.


So the perpendicular slope is 3%2F7



So now we know the slope of the unknown line is 3%2F7 (its the negative reciprocal of -7%2F3 from the line y=%28-7%2F3%29%2Ax-9). Also since the unknown line goes through (2,-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=%283%2F7%29%2A%28x-2%29 Plug in m=3%2F7, x%5B1%5D=2, and y%5B1%5D=-4



y%2B4=%283%2F7%29%2Ax-%283%2F7%29%282%29 Distribute 3%2F7



y%2B4=%283%2F7%29%2Ax-6%2F7 Multiply



y=%283%2F7%29%2Ax-6%2F7-4Subtract -4 from both sides to isolate y

y=%283%2F7%29%2Ax-6%2F7-28%2F7 Make into equivalent fractions with equal denominators



y=%283%2F7%29%2Ax-34%2F7 Combine the fractions



y=%283%2F7%29%2Ax-34%2F7 Reduce any fractions

So the equation of the line that is perpendicular to y=%28-7%2F3%29%2Ax-9 and goes through (2,-4) is y=%283%2F7%29%2Ax-34%2F7


So here are the graphs of the equations y=%28-7%2F3%29%2Ax-9 and y=%283%2F7%29%2Ax-34%2F7




graph of the given equation y=%28-7%2F3%29%2Ax-9 (red) and graph of the line y=%283%2F7%29%2Ax-34%2F7(green) that is perpendicular to the given graph and goes through (2,-4)