SOLUTION: Find the equation of the line through (4,3) perpendicular to the line 3x + y =7 Show all work.

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Question 88837This question is from textbook algebra and trigonometry
: Find the equation of the line through (4,3) perpendicular to the line 3x + y =7
Show all work.
 This question is from textbook algebra and trigonometry

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First convert 3x+%2B+y+=7 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)


3x%2B1y=7 Start with the given equation


3x%2B1y-3x=7-3x Subtract 3x from both sides


1y=-3x%2B7 Simplify


The original equation 3x%2B1y=7 (standard form) is equivalent to y+=+-3x%2B7 (slope-intercept form)


The equation y+=+-3x%2B7 is in the form y=mx%2Bb where m=-3 is the slope and b=7 is the y intercept.






Now lets find the line perpendicular to y=-3x%2B7 through (4,3)


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%2B7). Also since the unknown line goes through (4,3), 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-3=%281%2F3%29%2A%28x-4%29 Plug in m=1%2F3, x%5B1%5D=4, and y%5B1%5D=3



y-3=%281%2F3%29%2Ax-%281%2F3%29%284%29 Distribute 1%2F3



y-3=%281%2F3%29%2Ax-4%2F3 Multiply



y=%281%2F3%29%2Ax-4%2F3%2B3Add 3 to both sides to isolate y

y=%281%2F3%29%2Ax-4%2F3%2B9%2F3 Make into equivalent fractions with equal denominators



y=%281%2F3%29%2Ax%2B5%2F3 Combine the fractions



y=%281%2F3%29%2Ax%2B5%2F3 Reduce any fractions

So the equation of the line that is perpendicular to y=-3%2Ax%2B7 and goes through (4,3) is y=%281%2F3%29%2Ax%2B5%2F3


So here are the graphs of the equations y=-3%2Ax%2B7 and y=%281%2F3%29%2Ax%2B5%2F3




graph of the given equation y=-3%2Ax%2B7 (red) and graph of the line y=%281%2F3%29%2Ax%2B5%2F3(green) that is perpendicular to the given graph and goes through (4,3)