SOLUTION: write an equation of the line containing a given point and perpendicular to the given line? (7,-6);7x+3y=5

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Question 171784: write an equation of the line containing a given point and perpendicular to the given line?
(7,-6);7x+3y=5
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

7x%2B3y=5 Start with the given equation.


3y=5-7x Subtract 7x from both sides.


3y=-7x%2B5 Rearrange the terms.


y=%28-7x%2B5%29%2F%283%29 Divide both sides by 3 to isolate y.


y=%28%28-7%29%2F%283%29%29x%2B%285%29%2F%283%29 Break up the fraction.


y=-%287%2F3%29x%2B5%2F3 Reduce.


We can see that the equation y=-%287%2F3%29x%2B5%2F3 has a slope m=-7%2F3 and a y-intercept b=5%2F3.


Now to find the slope of the perpendicular line, simply flip the slope m=-7%2F3 to get m=-3%2F7. Now change the sign to get m=3%2F7. So the perpendicular slope is m=3%2F7.


Now let's use the point slope formula to find the equation of the perpendicular line by plugging in the slope m=-7%2F3 and the coordinates of the given point .


y-y%5B1%5D=m%28x-x%5B1%5D%29 Start with the point slope formula


y--6=%283%2F7%29%28x-7%29 Plug in m=3%2F7, x%5B1%5D=7, and y%5B1%5D=-6


y%2B6=%283%2F7%29%28x-7%29 Rewrite y--6 as y%2B6


y%2B6=%283%2F7%29x%2B%283%2F7%29%28-7%29 Distribute


y%2B6=%283%2F7%29x-3 Multiply


y=%283%2F7%29x-3-6 Subtract 6 from both sides.


y=%283%2F7%29x-9 Combine like terms.


So the equation of the line perpendicular to 7x%2B3y=5 that goes through the point is y=%283%2F7%29x-9.


Here's a graph to visually verify our answer:
Graph of the original equation y=-%287%2F3%29x%2B5%2F3 (red) and the perpendicular line y=%283%2F7%29x-9 (green) through the point .