SOLUTION: Write an equation of a line passing thru (-4,2) perpendicular to -6x+5y=16 How do I set this problem up?

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Question 92089: Write an equation of a line passing thru (-4,2) perpendicular to -6x+5y=16
How do I set this problem up?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First convert -6x+5y=16 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)


-6x%2B5y=16 Start with the given equation


-6x%2B5y%2B6x=16%2B6x Add 6x to both sides


5y=6x%2B16 Simplify


%285y%29%2F%285%29=%286x%2B16%29%2F%285%29 Divide both sides by 5 to isolate y


y+=+%286x%29%2F%285%29%2B%2816%29%2F%285%29 Break up the fraction on the right hand side


y+=+%286%2F5%29x%2B16%2F5 Reduce and simplify


The original equation -6x%2B5y=16 (standard form) is equivalent to y+=+%286%2F5%29x%2B16%2F5 (slope-intercept form)


The equation y+=+%286%2F5%29x%2B16%2F5 is in the form y=mx%2Bb where m=6%2F5 is the slope and b=16%2F5 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 6%2F5, 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%286%2F5%29 So plug in the given slope to find the perpendicular slope



m%5Bp%5D=%28-1%2F1%29%285%2F6%29 When you divide fractions, you multiply the first fraction (which is really 1%2F1) by the reciprocal of the second



m%5Bp%5D=-5%2F6 Multiply the fractions.


So the perpendicular slope is -5%2F6



So now we know the slope of the unknown line is -5%2F6 (its the negative reciprocal of 6%2F5 from the line y=%286%2F5%29%2Ax%2B16%2F5). Also since the unknown line goes through (-4,2), 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-2=%28-5%2F6%29%2A%28x%2B4%29 Plug in m=-5%2F6, x%5B1%5D=-4, and y%5B1%5D=2



y-2=%28-5%2F6%29%2Ax%2B%285%2F6%29%28-4%29 Distribute -5%2F6



y-2=%28-5%2F6%29%2Ax-20%2F6 Multiply



y=%28-5%2F6%29%2Ax-20%2F6%2B2Add 2 to both sides to isolate y

y=%28-5%2F6%29%2Ax-20%2F6%2B12%2F6 Make into equivalent fractions with equal denominators



y=%28-5%2F6%29%2Ax-8%2F6 Combine the fractions



y=%28-5%2F6%29%2Ax-4%2F3 Reduce any fractions

So the equation of the line that is perpendicular to y=%286%2F5%29%2Ax%2B16%2F5 and goes through (-4,2) is y=%28-5%2F6%29%2Ax-4%2F3


So here are the graphs of the equations y=%286%2F5%29%2Ax%2B16%2F5 and y=%28-5%2F6%29%2Ax-4%2F3




graph of the given equation y=%286%2F5%29%2Ax%2B16%2F5 (red) and graph of the line y=%28-5%2F6%29%2Ax-4%2F3(green) that is perpendicular to the given graph and goes through (-4,2)