SOLUTION: Find the equation, in standard form, with all interger coeffecients, of the line perpendicular to 3x - 6y = 9 and passing through (-2,-1)

Algebra ->  Coordinate-system -> SOLUTION: Find the equation, in standard form, with all interger coeffecients, of the line perpendicular to 3x - 6y = 9 and passing through (-2,-1)      Log On


   

Question 114588: Find the equation, in standard form, with all interger coeffecients, of the line perpendicular to 3x - 6y = 9 and passing through (-2,-1)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First convert 3x - 6y = 9 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)


3x-6y=9 Start with the given equation


3x-6y-3x=9-3x Subtract 3x from both sides


-6y=-3x%2B9 Simplify


%28-6y%29%2F%28-6%29=%28-3x%2B9%29%2F%28-6%29 Divide both sides by -6 to isolate y


y+=+%28-3x%29%2F%28-6%29%2B%289%29%2F%28-6%29 Break up the fraction on the right hand side


y+=+%281%2F2%29x-3%2F2 Reduce and simplify


The original equation 3x-6y=9 (standard form) is equivalent to y+=+%281%2F2%29x-3%2F2 (slope-intercept form)


The equation y+=+%281%2F2%29x-3%2F2 is in the form y=mx%2Bb where m=1%2F2 is the slope and b=-3%2F2 is the y intercept.





Now let's find the equation of the line through points (-2,-1) that is perpendicular to y=%281%2F2%29x-3%2F2
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 1%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%281%2F2%29 So plug in the given slope to find the perpendicular slope



m%5Bp%5D=%28-1%2F1%29%282%2F1%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%2F1 Multiply the fractions.


So the perpendicular slope is -2



So now we know the slope of the unknown line is -2 (its the negative reciprocal of 1%2F2 from the line y=%281%2F2%29%2Ax-3%2F2). Also since the unknown line goes through (-2,-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%2B1=-2%2A%28x%2B2%29 Plug in m=-2, x%5B1%5D=-2, and y%5B1%5D=-1



y%2B1=-2%2Ax%2B%282%29%28-2%29 Distribute -2



y%2B1=-2%2Ax-4 Multiply



y=-2%2Ax-4-1Subtract -1 from both sides to isolate y

y=-2%2Ax-5 Combine like terms

So the equation of the line that is perpendicular to y=%281%2F2%29%2Ax-3%2F2 and goes through (-2,-1) is y=-2%2Ax-5


So here are the graphs of the equations y=%281%2F2%29%2Ax-3%2F2 and y=-2%2Ax-5




graph of the given equation y=%281%2F2%29%2Ax-3%2F2 (red) and graph of the line y=-2%2Ax-5(green) that is perpendicular to the given graph and goes through (-2,-1)