SOLUTION: find the equation of the line through point (1,1) which is perpendicular to the line 2x-3y=12

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Question 98543: find the equation of the line through point (1,1) which is perpendicular to the line 2x-3y=12
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First convert 2x-3y=12 to 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)


2x-3y=12 Start with the given equation


2x-3y-2x=12-2x Subtract 2x from both sides


-3y=-2x%2B12 Simplify


%28-3y%29%2F%28-3%29=%28-2x%2B12%29%2F%28-3%29 Divide both sides by -3 to isolate y


y+=+%28-2x%29%2F%28-3%29%2B%2812%29%2F%28-3%29 Break up the fraction on the right hand side


y+=+%282%2F3%29x-4 Reduce and simplify


The original equation 2x-3y=12 (standard form) is equivalent to y+=+%282%2F3%29x-4 (slope-intercept form)


The equation y+=+%282%2F3%29x-4 is in the form y=mx%2Bb where m=2%2F3 is the slope and b=-4 is the y intercept.





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-4). Also since the unknown line goes through (1,1), 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-1=%28-3%2F2%29%2A%28x-1%29 Plug in m=-3%2F2, x%5B1%5D=1, and y%5B1%5D=1



y-1=%28-3%2F2%29%2Ax%2B%283%2F2%29%281%29 Distribute -3%2F2



y-1=%28-3%2F2%29%2Ax%2B3%2F2 Multiply



y=%28-3%2F2%29%2Ax%2B3%2F2%2B1Add 1 to both sides to isolate y

y=%28-3%2F2%29%2Ax%2B3%2F2%2B2%2F2 Make into equivalent fractions with equal denominators



y=%28-3%2F2%29%2Ax%2B5%2F2 Combine the fractions



y=%28-3%2F2%29%2Ax%2B5%2F2 Reduce any fractions

So the equation of the line that is perpendicular to y=%282%2F3%29%2Ax-4 and goes through (1,1) is y=%28-3%2F2%29%2Ax%2B5%2F2


So here are the graphs of the equations y=%282%2F3%29%2Ax-4 and y=%28-3%2F2%29%2Ax%2B5%2F2




graph of the given equation y=%282%2F3%29%2Ax-4 (red) and graph of the line y=%28-3%2F2%29%2Ax%2B5%2F2(green) that is perpendicular to the given graph and goes through (1,1)