SOLUTION: Find the equation, in standard form, with all integer coefficients, of the line perpendicular to x + 3y = 12 and passing through (9, -5). I normally do not ask for help, but I

Algebra ->  Graphs -> SOLUTION: Find the equation, in standard form, with all integer coefficients, of the line perpendicular to x + 3y = 12 and passing through (9, -5). I normally do not ask for help, but I      Log On


   

Question 104002: Find the equation, in standard form, with all integer coefficients, of the line perpendicular to x + 3y = 12 and passing through (9, -5).

I normally do not ask for help, but I am lost on this problem. If anyone can help, I would greatly appreciate it.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

First convert the standard equation x%2B3y=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)


1x%2B3y=12 Start with the given equation


1x%2B3y-1x=12-1x Subtract 1x from both sides


3y=-1x%2B12 Simplify


%283y%29%2F%283%29=%28-1x%2B12%29%2F%283%29 Divide both sides by 3 to isolate y


y+=+%28-1x%29%2F%283%29%2B%2812%29%2F%283%29 Break up the fraction on the right hand side


y+=+%28-1%2F3%29x%2B4 Reduce and simplify


The original equation 1x%2B3y=12 (standard form) is equivalent to y+=+%28-1%2F3%29x%2B4 (slope-intercept form)


The equation y+=+%28-1%2F3%29x%2B4 is in the form y=mx%2Bb where m=-1%2F3 is the slope and b=4 is the y intercept.







Now let's find the equation of the line that is perpendicular to y=%28-1%2F3%29x%2B4 which goes through (9,-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 -1%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%28-1%2F3%29 So plug in the given slope to find the perpendicular slope



m%5Bp%5D=%28-1%2F1%29%283%2F-1%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%2F1 Multiply the fractions.


So the perpendicular slope is 3



So now we know the slope of the unknown line is 3 (its the negative reciprocal of -1%2F3 from the line y=%28-1%2F3%29%2Ax%2B4). Also since the unknown line goes through (9,-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%2B5=3%2A%28x-9%29 Plug in m=3, x%5B1%5D=9, and y%5B1%5D=-5



y%2B5=3%2Ax-%283%29%289%29 Distribute 3



y%2B5=3%2Ax-27 Multiply



y=3%2Ax-27-5Subtract -5 from both sides to isolate y

y=3%2Ax-32 Combine like terms

So the equation of the line that is perpendicular to y=%28-1%2F3%29%2Ax%2B4 and goes through (9,-5) is y=3%2Ax-32


So here are the graphs of the equations y=%28-1%2F3%29%2Ax%2B4 and y=3%2Ax-32




graph of the given equation y=%28-1%2F3%29%2Ax%2B4 (red) and graph of the line y=3%2Ax-32(green) that is perpendicular to the given graph and goes through (9,-5)





Now convert y=3x-32 into standard form


Solved by pluggable solver: Converting Linear Equations in Standard form to Slope-Intercept Form (and vice versa)
Convert from slope-intercept form (y = mx+b) to standard form (Ax+By = C)


y+=+3x-32 Start with the given equation


1y-3x+=+3x-32-3x Subtract 3x from both sides


-3x%2B1y+=+-32 Simplify


-1%2A%28-3x%2B1y%29+=+-1%2A%28-32%29 Multiply both sides by -1 to make the A coefficient positive (note: this step may be optional; it will depend on your teacher and/or textbook)


3x-1y+=+32 Distribute and simplify


The original equation y+=+3x-32 (slope-intercept form) is equivalent to 3x-1y+=+32 (standard form where A > 0)


The equation 3x-1y+=+32 is in the form Ax%2BBy+=+C where A+=+3, B+=+-1 and C+=+32






So the equation of the line that is perpendicular to x + 3y = 12 and passing through (9, -5) is


-3x%2By=-32