SOLUTION: equation of the line through points (6,-2) and perpendicular to the line 2x-3y=-7

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Question 114572: equation of the line through points (6,-2) and perpendicular to the line 2x-3y=-7
Found 2 solutions by jim_thompson5910, checkley71:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First convert 2x-3y=-7 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=-7 Start with the given equation


2x-3y-2x=-7-2x Subtract 2x from both sides


-3y=-2x-7 Simplify


%28-3y%29%2F%28-3%29=%28-2x-7%29%2F%28-3%29 Divide both sides by -3 to isolate y


y+=+%28-2x%29%2F%28-3%29%2B%28-7%29%2F%28-3%29 Break up the fraction on the right hand side


y+=+%282%2F3%29x%2B7%2F3 Reduce and simplify


The original equation 2x-3y=-7 (standard form) is equivalent to y+=+%282%2F3%29x%2B7%2F3 (slope-intercept form)


The equation y+=+%282%2F3%29x%2B7%2F3 is in the form y=mx%2Bb where m=2%2F3 is the slope and b=7%2F3 is the y intercept.





Now let's find the equation of the line through points (6,-2) that is perpendicular to y=%282%2F3%29x%2B7%2F3
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%2B7%2F3). Also since the unknown line goes through (6,-2), 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%2B2=%28-3%2F2%29%2A%28x-6%29 Plug in m=-3%2F2, x%5B1%5D=6, and y%5B1%5D=-2



y%2B2=%28-3%2F2%29%2Ax%2B%283%2F2%29%286%29 Distribute -3%2F2



y%2B2=%28-3%2F2%29%2Ax%2B18%2F2 Multiply



y=%28-3%2F2%29%2Ax%2B18%2F2-2Subtract -2 from both sides to isolate y

y=%28-3%2F2%29%2Ax%2B18%2F2-4%2F2 Make into equivalent fractions with equal denominators



y=%28-3%2F2%29%2Ax%2B14%2F2 Combine the fractions



y=%28-3%2F2%29%2Ax%2B7 Reduce any fractions

So the equation of the line that is perpendicular to y=%282%2F3%29%2Ax%2B7%2F3 and goes through (6,-2) is y=%28-3%2F2%29%2Ax%2B7


So here are the graphs of the equations y=%282%2F3%29%2Ax%2B7%2F3 and y=%28-3%2F2%29%2Ax%2B7




graph of the given equation y=%282%2F3%29%2Ax%2B7%2F3 (red) and graph of the line y=%28-3%2F2%29%2Ax%2B7(green) that is perpendicular to the given graph and goes through (6,-2)

Answer by checkley71(8403) About Me  (Show Source):
You can put this solution on YOUR website!
FIRST YOU NEED TO FIND THE SLOPE OF THE GIVEN LINE.
Y=mX+b WHERE (m)=SLOPE.
2X-3Y=-7
-3Y=-2X-7
Y=-2X/-3-7/-3
Y=2X/3+7/3 THUS THIS SLOPE=2/3.
A PERPENDICULAR LINE WOULD HAVE A SLOPE = TO THE NEGATIVE RECIPRICAL OF THIS SLOPE.
THUS THE REQUIRED SLOPE=-3/2 FOR THE PERPENDICULAR LINE.
USING THE LINE FORMULA Y=mX+b & SUBSTITUTING THE POINT (6,-2) & THE SLOPE=-3/2 YOU NEED TO SOLVE FOR b THE Y INTERCEPT.
-2=-3/2*6+b
-2=-18/2+b
b=-2+9
b=7
SO THE LINE EQUATION IS:
Y=-3X/2+7