Expert: Bobby Soltani - 7/29/2004

Question
The equation : L=5x+6y=9

a) Find the equation of the line parallel L through the point (2,3)

b) Find the equation of the line perpendicular to L through (4,-8)

Please explain the answer and show how to do it.

From Samson

Answer
Hi Samson,

In order to find the equation for a line, we need a point on the line and the slope.  For part a and b, we are given a point but we need to find the correct slope.  We are given that the desired lines should be parallel (a) to L and perpendicular (b) to L.  Two lines that are parallel will have the same slope and two lines that are perpendicular will have slopes that are negative recipricals of each other.  For instance, if the slope of one line is 3, then the line perpendicular to it would have a slope of -1/3.  To find the slope of the line L we need to get it into the for y = mx + b.  Then, m will be the slope.  I'll do that here.
5x + 6y = 9
6y = -5x + 9              (subtracted 5x from each side)
y = (-5/6)x + (9/6)       (divided both sides by 6)

So, the slope of the line is -5/6.  So the line in part a should have a slope of m = -5/6.  The line in part b should have a slope of m = 6/5.

To find the equation of a line using one point and a slope, we use the following form: y - y1 = m*(x - x1)
x1 and y1 are the coordinates of the given point and m is the slope.

part a) x1 = 2, y1 = 3, m = -5/6
plug it in to our equation
y - 3 = (-5/6)*(x - 2)
y - 3 = (-5/6)x + 5/3

y = (-5/6)*x + 14/3
That is the answer to part a.

Do part b the same with x1 = 4, y1 = -8, and m = 6/5

Let me know if you have any more questions.  I hope this helps you out.  Good Luck.

Bobby