SOLUTION: Write the equation that represents a line passing through (0,5) and is perpendicular to 4x+6y=12

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Question 102591: Write the equation that represents a line passing through (0,5) and is perpendicular to 4x+6y=12
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!



First convert the standard equation 4x%2B6y=12 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)


4x%2B6y=12 Start with the given equation


4x%2B6y-4x=12-4x Subtract 4x from both sides


6y=-4x%2B12 Simplify


%286y%29%2F%286%29=%28-4x%2B12%29%2F%286%29 Divide both sides by 6 to isolate y


y+=+%28-4x%29%2F%286%29%2B%2812%29%2F%286%29 Break up the fraction on the right hand side


y+=+%28-2%2F3%29x%2B2 Reduce and simplify


The original equation 4x%2B6y=12 (standard form) is equivalent to y+=+%28-2%2F3%29x%2B2 (slope-intercept form)


The equation y+=+%28-2%2F3%29x%2B2 is in the form y=mx%2Bb where m=-2%2F3 is the slope and b=2 is the y intercept.







Now let's find the equation of the line that is perpendicular to y=%28-2%2F3%29x%2B2 which goes through (0,5)

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



m%5Bp%5D=%28-1%2F1%29%283%2F-2%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=%28-2%2F3%29%2Ax%2B2). Also since the unknown line goes through (0,5), 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=%283%2F2%29%2A%28x-0%29 Plug in m=3%2F2, x%5B1%5D=0, and y%5B1%5D=5



y-5=%283%2F2%29%2Ax-%283%2F2%29%280%29 Distribute 3%2F2



y-5=%283%2F2%29%2Ax-0%2F2 Multiply



y=%283%2F2%29%2Ax-0%2F2%2B5Add 5 to both sides to isolate y

y=%283%2F2%29%2Ax-0%2F2%2B10%2F2 Make into equivalent fractions with equal denominators



y=%283%2F2%29%2Ax%2B10%2F2 Combine the fractions



y=%283%2F2%29%2Ax%2B5 Reduce any fractions

So the equation of the line that is perpendicular to y=%28-2%2F3%29%2Ax%2B2 and goes through (0,5) is y=%283%2F2%29%2Ax%2B5


So here are the graphs of the equations y=%28-2%2F3%29%2Ax%2B2 and y=%283%2F2%29%2Ax%2B5




graph of the given equation y=%28-2%2F3%29%2Ax%2B2 (red) and graph of the line y=%283%2F2%29%2Ax%2B5(green) that is perpendicular to the given graph and goes through (0,5)