SOLUTION: Write the slope-intercept form of the equation of a line passing through the point (-6,5) and perpendicular to the line 3x-2y=8

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Question 99558: Write the slope-intercept form of the equation of a line passing through the point (-6,5) and perpendicular to the line 3x-2y=8
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!



First convert the standard equation 3x-2y=8 into slope intercept form

Solved by pluggable solver: Converting Linear Equations in Standard form to Slope-Intercept Form (and vice versa)
Convert from standard form (Ax+By = C) to slope-intercept form (y = mx+b)


3x-2y=8 Start with the given equation


3x-2y-3x=8-3x Subtract 3x from both sides


-2y=-3x%2B8 Simplify


%28-2y%29%2F%28-2%29=%28-3x%2B8%29%2F%28-2%29 Divide both sides by -2 to isolate y


y+=+%28-3x%29%2F%28-2%29%2B%288%29%2F%28-2%29 Break up the fraction on the right hand side


y+=+%283%2F2%29x-4 Reduce and simplify


The original equation 3x-2y=8 (standard form) is equivalent to y+=+%283%2F2%29x-4 (slope-intercept form)


The equation y+=+%283%2F2%29x-4 is in the form y=mx%2Bb where m=3%2F2 is the slope and b=-4 is the y intercept.







Now let's find the equation of the line that is perpendicular to y=%283%2F2%29x-4 which goes through (-6,5)

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-4). Also since the unknown line goes through (-6,5), 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-5=%28-2%2F3%29%2A%28x%2B6%29 Plug in m=-2%2F3, x%5B1%5D=-6, and y%5B1%5D=5



y-5=%28-2%2F3%29%2Ax%2B%282%2F3%29%28-6%29 Distribute -2%2F3



y-5=%28-2%2F3%29%2Ax-12%2F3 Multiply



y=%28-2%2F3%29%2Ax-12%2F3%2B5Add 5 to both sides to isolate y

y=%28-2%2F3%29%2Ax-12%2F3%2B15%2F3 Make into equivalent fractions with equal denominators



y=%28-2%2F3%29%2Ax%2B3%2F3 Combine the fractions



y=%28-2%2F3%29%2Ax%2B1 Reduce any fractions

So the equation of the line that is perpendicular to y=%283%2F2%29%2Ax-4 and goes through (-6,5) is y=%28-2%2F3%29%2Ax%2B1


So here are the graphs of the equations y=%283%2F2%29%2Ax-4 and y=%28-2%2F3%29%2Ax%2B1




graph of the given equation y=%283%2F2%29%2Ax-4 (red) and graph of the line y=%28-2%2F3%29%2Ax%2B1(green) that is perpendicular to the given graph and goes through (-6,5)