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# Equations of Parallel Lines

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.

Method 1: Find the gradient of the given line, and that should be the gradient of the required equation of the line. The use point gradient form to find the equation of the line.

2x + 3y + 6 = 0

2x + 3y = -6

3y = -2x – 6

y = y = 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