When two lines are parallel, they will make the same angle with x-axis, hence their gradients are the same.
In other words, if a line with gradient m1 is parallel to a line with gradient m2, then m1 = m2.
Hence any two equations of parallel lines (expressed in general form or otherwise) will only have the constant part different i.e.
Ax + By + C1 = 0, and
Ax + By + C2 = 0 are parallel lines.
Note that the only change in the equations of the parallel lines above is the constants C1 and C2.
For example, 2x + 3y + 6 = 0 and 2x + 3y – 4 = 0 are parallel lines.
In general, a set of parallel lines can be given by ax + by + k = 0 where k can take any real value.
Example 1: Find the equation of the straight line that is parallel to 2x + 3y + 6 = 0 and that passes through (2, 3).
We can solve this problem in 2 methods.
2x + 3y + 6 = 0
2x + 3y = -6
3y = -2x – 6
So m = y = is the required gradient of the line.
This line passes through the point (2, 3). Using point gradient form, we get
y – y1 = m (x – x1)
y – 3 = (x – 2)
Multiplying by 3 on both sides –
3y – 9 = -2(x – 2)
3y – 9 = -2x + 4
2x + 3y – 4 – 9 = 0
2x + 3y – 13 = 0 is the equation of the parallel line
Method 2: Use the definition of a parallel line. Substitute the given point to get the constant, and the required equation of the parallel line.
Our given equation is 2x + 3y + 6 = 0.
Equation of a parallel line is given by ax + by + k = 0
Substituting (2, 3) in the above equation, we get –
2 x 2 + 3 x 3 + k = 0
4 + 9 + k = 0
k = -13
So the required equation of the parallel line is 2x + 3y – 13 = 0