# Equation of Line Parallel to a Line

Here you will learn what is the equation of line parallel to a given line with examples.

Let’s begin –

## Equation of Line Parallel to a Line

The equation of the line parallel to a given line ax + by + c = 0 is

ax + by + $$\lambda$$,

where $$\lambda$$ is a constant.

Proof :

Let m be the slope of the line ax + by + c = 0, Then,

m = -$$a\over b$$

Since the required line is parallel to the given line, the slope of the required line is also m.

Let $$c_1$$ be the y-intercept of the required line. Then, its equation is

y = mx + $$c_1$$

y = -$$a\over b$$x + $$c_1$$

$$\implies$$ ax + by – b$$c_1$$ = 0

$$\implies$$ ax + by + $$\lambda$$ = 0, where $$\lambda$$ = -b$$c_1$$ = constant.

Note : To write a line parallel to any given line we keep the expression containing x and y same and simply replace the given constant by a new constant $$\lambda$$. The value of $$\lambda$$ can be determined by some given condition.

Example : Find the equation of line which is parallal to the line 3x – 2y + 5 = 0 and passes through the point (5, -6).

Solution :  The line parallel to the line 3x – 2y + 5 = 0 is

3x – 3y + $$\lambda$$ = 0                    …………..(i)

This passes through (5, -6)

$$\therefore$$ 3 $$\times$$ 5 – 2 $$\times$$ -6 + $$\lambda$$ = 0

$$\implies$$ $$\lambda$$ = -27.

Putting $$\lambda$$ = -27 in  (i) we get,

3x – 3y – 27 = 0, which is the required equation of line.