SOLUTION: I have two questions on the equations of lines. 1)Write an equation of a line passing through the points (-3,5) and (2,-7) 2) Write an equation of the line perpendicular to

Algebra ->  Rational-functions -> SOLUTION: I have two questions on the equations of lines. 1)Write an equation of a line passing through the points (-3,5) and (2,-7) 2) Write an equation of the line perpendicular to      Log On


   

Question 275270: I have two questions on the equations of lines.
1)Write an equation of a line passing through the points (-3,5) and (2,-7)

2) Write an equation of the line perpendicular to the line 2x-4y=7 and passing through (2,-5)
I am slightly perplexed, thanks in advance.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
# 1



First let's find the slope of the line through the points and


Note: is the first point . So this means that x%5B1%5D=-3 and y%5B1%5D=5.
Also, is the second point . So this means that x%5B2%5D=2 and y%5B2%5D=-7.


m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula.


m=%28-7-5%29%2F%282--3%29 Plug in y%5B2%5D=-7, y%5B1%5D=5, x%5B2%5D=2, and x%5B1%5D=-3


m=%28-12%29%2F%282--3%29 Subtract 5 from -7 to get -12


m=%28-12%29%2F%285%29 Subtract -3 from 2 to get 5


So the slope of the line that goes through the points and is m=-12%2F5


Now let's use the point slope formula:


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


y-5=%28-12%2F5%29%28x--3%29 Plug in m=-12%2F5, x%5B1%5D=-3, and y%5B1%5D=5


y-5=%28-12%2F5%29%28x%2B3%29 Rewrite x--3 as x%2B3


y-5=%28-12%2F5%29x%2B%28-12%2F5%29%283%29 Distribute


y-5=%28-12%2F5%29x-36%2F5 Multiply


y=%28-12%2F5%29x-36%2F5%2B5 Add 5 to both sides.


y=%28-12%2F5%29x-11%2F5 Combine like terms. note: If you need help with fractions, check out this solver.


So the equation that goes through the points and is y=%28-12%2F5%29x-11%2F5


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# 2


2x-4y=7 Start with the given equation.


-4y=7-2x Subtract 2x from both sides.


-4y=-2x%2B7 Rearrange the terms.


y=%28-2x%2B7%29%2F%28-4%29 Divide both sides by -4 to isolate y.


y=%28%28-2%29%2F%28-4%29%29x%2B%287%29%2F%28-4%29 Break up the fraction.


y=%281%2F2%29x-7%2F4 Reduce.


So the slope of y=%281%2F2%29x-7%2F4 and 2x-4y=7 is 1%2F2

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%2F2, 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%281%2F2%29 So plug in the given slope to find the perpendicular slope



m%5Bp%5D=%28-1%2F1%29%282%2F1%29 When you divide fractions, you multiply the first fraction (which is really 1%2F1) by the reciprocal of the second



m%5Bp%5D=-2%2F1 Multiply the fractions.


So the perpendicular slope is -2



So now we know the slope of the unknown line is -2 (its the negative reciprocal of 1%2F2 from the line y=%281%2F2%29%2Ax-7%2F4). Also since the unknown line goes through (2,-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=-2%2A%28x-2%29 Plug in m=-2, x%5B1%5D=2, and y%5B1%5D=-5



y%2B5=-2%2Ax%2B%282%29%282%29 Distribute -2



y%2B5=-2%2Ax%2B4 Multiply



y=-2%2Ax%2B4-5Subtract -5 from both sides to isolate y

y=-2%2Ax-1 Combine like terms

So the equation of the line that is perpendicular to y=%281%2F2%29%2Ax-7%2F4 and goes through (2,-5) is y=-2%2Ax-1


So here are the graphs of the equations y=%281%2F2%29%2Ax-7%2F4 and y=-2%2Ax-1




graph of the given equation y=%281%2F2%29%2Ax-7%2F4 (red) and graph of the line y=-2%2Ax-1(green) that is perpendicular to the given graph and goes through (2,-5)