SOLUTION: Write the equation of each line. Give the answer in standard formusing only integers as the coefficients. The line through (2,-3) that is perpendicular to the line y=-3x+12 The

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Question 118012: Write the equation of each line. Give the answer in standard formusing only integers as the coefficients.
The line through (2,-3) that is perpendicular to the line y=-3x+12
The answer is x-3y=11
How do I work the problem to find the answer given?
Answer by jim_thompson5910(28593) About Me  (Show Source):
You can put this solution on YOUR website!
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 -3, 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%28-3%2F1%29 So plug in the given slope to find the perpendicular slope



m%5Bp%5D=%28-1%2F1%29%281%2F-3%29 When you divide fractions, you multiply the first fraction (which is really 1%2F1) by the reciprocal of the second



m%5Bp%5D=1%2F3 Multiply the fractions.


So the perpendicular slope is 1%2F3



So now we know the slope of the unknown line is 1%2F3 (its the negative reciprocal of -3 from the line y=-3%2Ax%2B12). Also since the unknown line goes through (2,-3), 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%2B3=%281%2F3%29%2A%28x-2%29 Plug in m=1%2F3, x%5B1%5D=2, and y%5B1%5D=-3



y%2B3=%281%2F3%29%2Ax-%281%2F3%29%282%29 Distribute 1%2F3



y%2B3=%281%2F3%29%2Ax-2%2F3 Multiply



y=%281%2F3%29%2Ax-2%2F3-3Subtract -3 from both sides to isolate y

y=%281%2F3%29%2Ax-2%2F3-9%2F3 Make into equivalent fractions with equal denominators



y=%281%2F3%29%2Ax-11%2F3 Combine the fractions



y=%281%2F3%29%2Ax-11%2F3 Reduce any fractions

So the equation of the line that is perpendicular to y=-3%2Ax%2B12 and goes through (2,-3) is y=%281%2F3%29%2Ax-11%2F3


So here are the graphs of the equations y=-3%2Ax%2B12 and y=%281%2F3%29%2Ax-11%2F3




graph of the given equation y=-3%2Ax%2B12 (red) and graph of the line y=%281%2F3%29%2Ax-11%2F3(green) that is perpendicular to the given graph and goes through (2,-3)





Now let's convert y=%281%2F3%29x-11%2F3 to standard form

Solved by pluggable solver: Converting Linear Equations in Standard form to Slope-Intercept Form (and vice versa)


y=%281%2F3%29%2Ax-11%2F3Start with the given equation



y-%281%2F3%29%2Ax=-11%2F3Subtract %281%2F3%29%2Ax from both sides

-%281%2F3%29%2Ax%2By=-11%2F3 Rearrange

3%28-%281%2F3%29%2Ax%2By%29=3%28-11%2F3%29 Multiply both sides by 3



3%28-1%2F3%29x%2B3%2Ay=3%28-11%2F3%29 Distribute



-%283%2F3%29%2Ax%2B3%2Ay=-33%2F3 Multiply

-1%2Ax%2B3%2Ay=-11 Reduce

%2B11 Reduce2

Now the equation is in standard form Ax%2BBy=C where A=1, B=3, and C=%2B11




Since the answer is x-3y=11, I'm assuming that the book wants A to be positive.


-1%28-x%2B3y%29=%28-1%29%28-11%29 Multiply both sides of -x%2B3y=-11 by -1

x-3y=11 Distribute and multiply


So the answer is x-3y=11