SOLUTION: 1.)Find the equation of line passing through the point(2, -5) and parallel to the line 4x-y+3=0. 2.)Find the equation of line passing through the point(-4, 3) and perpendicular

Algebra ->  Linear-equations -> SOLUTION: 1.)Find the equation of line passing through the point(2, -5) and parallel to the line 4x-y+3=0. 2.)Find the equation of line passing through the point(-4, 3) and perpendicular       Log On


   

Question 477030: 1.)Find the equation of line passing through the point(2, -5) and parallel to the line 4x-y+3=0.
2.)Find the equation of line passing through the point(-4, 3) and perpendicular to the line 2x+6y-5=0.
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
1.)Find the equation of line passing through the point(2, -5) and parallel to the line 4x-y%2B3=0.
4x-y%2B3=0....->.....in slope-intercept form: 4x%2B3=y
Solved by pluggable solver: Finding the Equation of a Line Parallel or Perpendicular to a Given Line


Since any two parallel lines have the same slope we know the slope of the unknown line is 4 (its from the slope of y=4%2Ax%2B3 which is also 4). 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=4%2A%28x-2%29 Plug in m=4, x%5B1%5D=2, and y%5B1%5D=-5



y%2B5=4%2Ax-%284%29%282%29 Distribute 4



y%2B5=4%2Ax-8 Multiply



y=4%2Ax-8-5Subtract -5 from both sides to isolate y

y=4%2Ax-13 Combine like terms

So the equation of the line that is parallel to y=4%2Ax%2B3 and goes through (2,-5) is y=4%2Ax-13


So here are the graphs of the equations y=4%2Ax%2B3 and y=4%2Ax-13



graph of the given equation y=4%2Ax%2B3 (red) and graph of the line y=4%2Ax-13(green) that is parallel to the given graph and goes through (2,-5)





2.)Find the equation of line passing through the point(-4,+3) and perpendicular to the line 2x%2B6y-5=0.
2x%2B6y-5=0....->.....in slope-intercept form:6y=-2x%2B5...->...y=-%282%2F6%29x%2B5%2F6...->...y=-%281%2F3%29x%2B5%2F6
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 -4, 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%28-4%2F1%29 So plug in the given slope to find the perpendicular slope



m%5Bp%5D=%28-1%2F1%29%281%2F-4%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%2F4 Multiply the fractions.


So the perpendicular slope is 1%2F4



So now we know the slope of the unknown line is 1%2F4 (its the negative reciprocal of -4 from the line y=-4%2Ax%2B3). Also since the unknown line goes through (1/3,5/6), 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-5%2F6=%281%2F4%29%2A%28x-1%2F3%29 Plug in m=1%2F4, x%5B1%5D=1%2F3, and y%5B1%5D=5%2F6



y-5%2F6=%281%2F4%29%2Ax-%281%2F4%29%281%2F3%29 Distribute 1%2F4



y-5%2F6=%281%2F4%29%2Ax-1%2F12 Multiply



y=%281%2F4%29%2Ax-1%2F12%2B5%2F6Add 5%2F6 to both sides to isolate y

y=%281%2F4%29%2Ax-1%2F12%2B10%2F12 Make into equivalent fractions with equal denominators



y=%281%2F4%29%2Ax%2B9%2F12 Combine the fractions



y=%281%2F4%29%2Ax%2B3%2F4 Reduce any fractions

So the equation of the line that is perpendicular to y=-4%2Ax%2B3 and goes through (1%2F3,5%2F6) is y=%281%2F4%29%2Ax%2B3%2F4


So here are the graphs of the equations y=-4%2Ax%2B3 and y=%281%2F4%29%2Ax%2B3%2F4




graph of the given equation y=-4%2Ax%2B3 (red) and graph of the line y=%281%2F4%29%2Ax%2B3%2F4(green) that is perpendicular to the given graph and goes through (1%2F3,5%2F6)