SOLUTION: write the equation of the line L. satisfying each of the following sets of geometric conditions. L passes through (5,6) and is perpendicular to 3x-5y=15 L has y-intercept(0,-3)

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Question 93842: write the equation of the line L. satisfying each of the following sets of geometric conditions. L passes through (5,6) and is perpendicular to 3x-5y=15
L has y-intercept(0,-3) and is paralell to -3x+5y=-15
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
"L passes through (5,6) and is perpendicular to 3x-5y=15"


First convert 3x-5y=15 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-5y=15 Start with the given equation


3x-5y-3x=15-3x Subtract 3x from both sides


-5y=-3x%2B15 Simplify


%28-5y%29%2F%28-5%29=%28-3x%2B15%29%2F%28-5%29 Divide both sides by -5 to isolate y


y+=+%28-3x%29%2F%28-5%29%2B%2815%29%2F%28-5%29 Break up the fraction on the right hand side


y+=+%283%2F5%29x-3 Reduce and simplify


The original equation 3x-5y=15 (standard form) is equivalent to y+=+%283%2F5%29x-3 (slope-intercept form)


The equation y+=+%283%2F5%29x-3 is in the form y=mx%2Bb where m=3%2F5 is the slope and b=-3 is the y intercept.





Now lets find the perpendicular line

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%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%283%2F5%29 So plug in the given slope to find the perpendicular slope



m%5Bp%5D=%28-1%2F1%29%285%2F3%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%2F3 Multiply the fractions.


So the perpendicular slope is -5%2F3



So now we know the slope of the unknown line is -5%2F3 (its the negative reciprocal of 3%2F5 from the line y=%283%2F5%29%2Ax-3). Also since the unknown line goes through (5,6), 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-6=%28-5%2F3%29%2A%28x-5%29 Plug in m=-5%2F3, x%5B1%5D=5, and y%5B1%5D=6



y-6=%28-5%2F3%29%2Ax%2B%285%2F3%29%285%29 Distribute -5%2F3



y-6=%28-5%2F3%29%2Ax%2B25%2F3 Multiply



y=%28-5%2F3%29%2Ax%2B25%2F3%2B6Add 6 to both sides to isolate y

y=%28-5%2F3%29%2Ax%2B25%2F3%2B18%2F3 Make into equivalent fractions with equal denominators



y=%28-5%2F3%29%2Ax%2B43%2F3 Combine the fractions



y=%28-5%2F3%29%2Ax%2B43%2F3 Reduce any fractions

So the equation of the line that is perpendicular to y=%283%2F5%29%2Ax-3 and goes through (5,6) is y=%28-5%2F3%29%2Ax%2B43%2F3


So here are the graphs of the equations y=%283%2F5%29%2Ax-3 and y=%28-5%2F3%29%2Ax%2B43%2F3




graph of the given equation y=%283%2F5%29%2Ax-3 (red) and graph of the line y=%28-5%2F3%29%2Ax%2B43%2F3(green) that is perpendicular to the given graph and goes through (5,6)








"L has y-intercept(0,-3) and is paralell to -3x+5y=-15"

First convert -3x+5y=-15 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%2B5y=-15 Start with the given equation


-3x%2B5y%2B3x=-15%2B3x Add 3x to both sides


5y=3x-15 Simplify


%285y%29%2F%285%29=%283x-15%29%2F%285%29 Divide both sides by 5 to isolate y


y+=+%283x%29%2F%285%29%2B%28-15%29%2F%285%29 Break up the fraction on the right hand side


y+=+%283%2F5%29x-3 Reduce and simplify


The original equation -3x%2B5y=-15 (standard form) is equivalent to y+=+%283%2F5%29x-3 (slope-intercept form)


The equation y+=+%283%2F5%29x-3 is in the form y=mx%2Bb where m=3%2F5 is the slope and b=-3 is the y intercept.





Now lets find the parallel line

Solved by pluggable solver: Finding the Equation of a Line Parallel or Perpendicular to a Given Line
The equation y=%283%2F5%29%2Ax-3 goes through the point (0,-3) (which is equal and parallel to the given equation)



So the equation parallel to -3x%2B5y=-15 and that goes through (0,-3) is %283%2F5%29x-3 (or -3x%2B5y=-15)