5.5 Parallel Lines

5.5 Parallel Lines

(A) Gradient of parallel lines

1. Two straight lines are 
    parallel if they have
    the same gradient.
    If PQ // RS,
    then mPQ = mRS
   
2. If two straight lines have 
    the same gradient, then  
    they are parallel. 
    If mAB = mCD
    then AB // CD
Example 1:
Determine whether the two straight lines are parallel.
(a) 2y – 4x = 6
      y = 2x 5
(b) 2y = 3x 4
      3y = 2x + 12

Solution:
(a) 
2y – 4x = 6
2y = 6 + 4x
y = 2x + 3,   m1 = 2
y = 2x 5,   m2 = 2
m1 = m2
Therefore, the two straight lines are parallel.

(b)
2y=3x4 y= 3 2 x2,    m 1 = 3 2 3y=2x+12 y= 2 3 x+4,    m 2 = 2 3 m 1 m 2  The two straight lines are not parallel.


(B) Equation of Parallel Lines

To find the equation of the straight line which passes through a given point and parallel to another straight line, follow the steps below:

Step 1: Let the equation of the straight line take the form y = mx + c.
Step 2: Find the gradient of the straight line from the equation of the 
            straight line parallel to it.
Step 3: Substitute the value of gradient, m, the x-coordinate and 
            y-coordinate of the given point into y = mx + c to find the value 
            of the y-intercept, c.
Step 4: Write down the equation of the straight line in the form
            y = mx + c.

Example 2:
Find the equation of the straight line that passes through the point (–8, 2) and is parallel to the straight line 4y + 3x = 12.

Solution:
4y+3x=12 4y=3x+12 y= 3 4 x+3 m= 3 4 At (8,2), substitute m= 3 4 , x=8y=2 into: y=mx+c 2= 3 4 ( 8 )+c c=26 c=4  The equation of the staright line is y= 3 4 x4.