SOLUTION: Please show all work including all steps so that I may understand, thank you!! Consider the point P = (2, 8) and the line L, given by the equation 6x - 3y = 15. A. Write an e

Algebra ->  Linear-equations -> SOLUTION: Please show all work including all steps so that I may understand, thank you!! Consider the point P = (2, 8) and the line L, given by the equation 6x - 3y = 15. A. Write an e      Log On


   

Question 938111: Please show all work including all steps so that I may understand, thank you!!
Consider the point P = (2, 8) and the line L, given by the equation 6x - 3y = 15.
A. Write an equation in slope-intercept of the line passing through P and parallel to line L.
B. Write an equation in slope-intercept of the line passing through P and perpendicular to line L.
Found 2 solutions by MathLover1, ewatrrr:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

the point P = (2, 8) and
the line L, given by the equation 6x+-3y+=+15
6x+-3y+=+15 in slope-intercept form 2x+-5+=+y
A. Write an equation in slope-intercept of the line passing through P and parallel to line L.

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 2 (its from the slope of y=2%2Ax-5 which is also 2). Also since the unknown line goes through (2,8), 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-8=2%2A%28x-2%29 Plug in m=2, x%5B1%5D=2, and y%5B1%5D=8



y-8=2%2Ax-%282%29%282%29 Distribute 2



y-8=2%2Ax-4 Multiply



y=2%2Ax-4%2B8Add 8 to both sides to isolate y

y=2%2Ax%2B4 Combine like terms

So the equation of the line that is parallel to y=2%2Ax-5 and goes through (2,8) is y=2%2Ax%2B4


So here are the graphs of the equations y=2%2Ax-5 and y=2%2Ax%2B4



graph of the given equation y=2%2Ax-5 (red) and graph of the line y=2%2Ax%2B4(green) that is parallel to the given graph and goes through (2,8)





B. Write an equation in slope-intercept of the line passing through P and perpendicular to line L.
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, 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%2F1%29 So plug in the given slope to find the perpendicular slope



m%5Bp%5D=%28-1%2F1%29%281%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=-1%2F2 Multiply the fractions.


So the perpendicular slope is -1%2F2



So now we know the slope of the unknown line is -1%2F2 (its the negative reciprocal of 2 from the line y=2%2Ax-5). Also since the unknown line goes through (2,8), 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-8=%28-1%2F2%29%2A%28x-2%29 Plug in m=-1%2F2, x%5B1%5D=2, and y%5B1%5D=8



y-8=%28-1%2F2%29%2Ax%2B%281%2F2%29%282%29 Distribute -1%2F2



y-8=%28-1%2F2%29%2Ax%2B2%2F2 Multiply



y=%28-1%2F2%29%2Ax%2B2%2F2%2B8Add 8 to both sides to isolate y

y=%28-1%2F2%29%2Ax%2B2%2F2%2B16%2F2 Make into equivalent fractions with equal denominators



y=%28-1%2F2%29%2Ax%2B18%2F2 Combine the fractions



y=%28-1%2F2%29%2Ax%2B9 Reduce any fractions

So the equation of the line that is perpendicular to y=2%2Ax-5 and goes through (2,8) is y=%28-1%2F2%29%2Ax%2B9


So here are the graphs of the equations y=2%2Ax-5 and y=%28-1%2F2%29%2Ax%2B9




graph of the given equation y=2%2Ax-5 (red) and graph of the line y=%28-1%2F2%29%2Ax%2B9(green) that is perpendicular to the given graph and goes through (2,8)

Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!
the standard slope-intercept form for an equation of a line is
y+=+highlight_green%28m%29x+%2B+highlight%28b%29
where m is the slope and b the y-intercept
...
the line L, given by the equation 6x - 3y = 15 0r y = 2x - 5(Green), m = 2
***Using point-slope form, y+-+y%5B1%5D+=+highlight_green%28m%29%28x+-+x%5B1%5D%29 P(2,8)
A. parallel to L, m = 2/1
y - 8 = 2(x-2)
y = 2x + 4 (Blue) 0r 2x-y = -4
....
B. perpendicular to L, m = -1/2 (negative reciprocal of 2 %282%2F1%29%28-1%2F2%29+=+-1
y - 8 = -1/2(x-2)
y = (-1/2)x + 9(Purple) 0r x + 2y = 18