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=3x−4 y= 3 2 x−2,    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=−8, y=2 into: y=mx+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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