SOLUTION: Determine the equation of the line. Perpendicular to 2x-3y+7=0 and with the same x-intercept as 4x+5y-8=0

Algebra ->  Coordinate Systems and Linear Equations  -> Lessons -> SOLUTION: Determine the equation of the line. Perpendicular to 2x-3y+7=0 and with the same x-intercept as 4x+5y-8=0      Log On


   

Question 934307: Determine the equation of the line. Perpendicular to 2x-3y+7=0 and with the same x-intercept as 4x+5y-8=0
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
the equation of the line perpendicular to 2x-3y%2B7=0 and with the same x-intercept as 4x%2B5y-8=0
first find x-intercept of 4x%2B5y-8=0:
4x%2B5y-8=0...set y=0 and solve for x
4x%2B5%2A0-8=0
4x=8
x=8%2F4
x=2
the x-intercept is at (2,0)
use this point and given line 2x-3y%2B7=0 in slope intercept form to find equation of perpendicular line which passes through (2,0):
2x%2B7=3y
y=%282%2F3%29x%2B7%2F3

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 2%2F3, 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%282%2F3%29 So plug in the given slope to find the perpendicular slope



m%5Bp%5D=%28-1%2F1%29%283%2F2%29 When you divide fractions, you multiply the first fraction (which is really 1%2F1) by the reciprocal of the second



m%5Bp%5D=-3%2F2 Multiply the fractions.


So the perpendicular slope is -3%2F2



So now we know the slope of the unknown line is -3%2F2 (its the negative reciprocal of 2%2F3 from the line y=%282%2F3%29%2Ax%2B7%2F3). Also since the unknown line goes through (2,0), 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-0=%28-3%2F2%29%2A%28x-2%29 Plug in m=-3%2F2, x%5B1%5D=2, and y%5B1%5D=0



y-0=%28-3%2F2%29%2Ax%2B%283%2F2%29%282%29 Distribute -3%2F2



y-0=%28-3%2F2%29%2Ax%2B6%2F2 Multiply



y=%28-3%2F2%29%2Ax%2B6%2F2%2B0Add 0 to both sides to isolate y

y=%28-3%2F2%29%2Ax%2B6%2F2%2B0%2F2 Make into equivalent fractions with equal denominators



y=%28-3%2F2%29%2Ax%2B6%2F2 Combine the fractions



y=%28-3%2F2%29%2Ax%2B3 Reduce any fractions

So the equation of the line that is perpendicular to y=%282%2F3%29%2Ax%2B7%2F3 and goes through (2,0) is y=%28-3%2F2%29%2Ax%2B3


So here are the graphs of the equations y=%282%2F3%29%2Ax%2B7%2F3 and y=%28-3%2F2%29%2Ax%2B3




graph of the given equation y=%282%2F3%29%2Ax%2B7%2F3 (red) and graph of the line y=%28-3%2F2%29%2Ax%2B3(green) that is perpendicular to the given graph and goes through (2,0)




so, the line y=-%283%2F2%29x%2B3(green) is perpendicular to line 2x-3y%2B7=0(red) and has same x-intercept as a line 4x%2B5y-8=0 (blue )
see all three on a graph: