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
= 2x + 3,   m1= 2
= 2x 5,   m2 = 2
m1= m2
Therefore, the two straight lines are parallel.
 
(b)
2 y = 3 x 4 y = 3 2 x 2 , m 1 = 3 2 3 y = 2 x + 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 + 3= 12.

Solution:
4 y + 3 x = 12 4 y = 3 x + 12 y = 3 4 x + 3 m = 3 4 At ( 8 , 2 ) , substitute m = 3 4 , x = 8 , y = 2 into: y = m x + c 2 = 3 4 ( 8 ) + c c = 2 6 c = 4 The equation of the staright line is y = 3 4 x 4.

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