# Condition of Parallelism

We will discuss here about the condition of parallelism.

If two lines are parallel then they are inclined at the same angle θ with the positive direction of the x-axis. So, their slopes are equal.

Two lines with slopes m$$_{1}$$ and m$$_{2}$$ are parallel if and only if m$$_{1}$$ = m$$_{2}$$

Note: If the slope of a line is m then any line parallel to it will also have the slope m.

Solved examples on condition of parallelism:

1. Prove that the lines 3x – 2y – 1 = 0 and 9x - 6y + 5 = 0 are parallel.

Solution:

The slope of the lines can be found by comparing the equations with y = mx + c.

Equation of the first straight line 3x – 2y – 1 = 0

Now we need to express the given equation in the form y = mx + c.

3x – 2y – 1 = 0

-2y = -3x + 1

y = $$\frac{-3}{-2}$$x + $$\frac{1}{-2}$$

y = $$\frac{3}{2}$$x - $$\frac{1}{2}$$

Therefore, the slope (m$$_{1}$$) of the given line = $$\frac{3}{2}$$

Equation of the second line 9x - 6y + 5 = 0

Now we need to express the given equation in the form y = mx + c.

9x - 6y + 5 = 0

-6y = -9x - 5

y = $$\frac{-9}{-6}$$x - $$\frac{5}{-6}$$

y = $$\frac{3}{2}$$x + $$\frac{5}{6}$$

Therefore, the slope (m$$_{2}$$) of the given line = $$\frac{3}{2}$$

Now we can clearly see that the slope of the first line m$$_{1}$$ = the slope of the second line m$$_{2}$$

Therefore, the given two lines are parallel.

2. Find the value of k if the lines 7y = kx + 4 and x + 2y = 3 are parallel.

Solution:

The slope of the lines can be found by comparing the equations with y = mx + c.

Equation of the first straight line 7y = kx + 4

Now we need to express the given equation in the form y = mx + c.

7y = kx + 4

y = $$\frac{k}{7}$$x + $$\frac{4}{7}$$

Therefore, the slope (m$$_{1}$$) of the given line = $$\frac{k}{7}$$

Equation of the second line x + 2y = 3

Now we need to express the given equation in the form y = mx + c.

x + 2y = 3

2y = -x + 3

y = -$$\frac{1}{2}$$x + $$\frac{3}{2}$$

Therefore, the slope (m$$_{2}$$) of the given line = -$$\frac{1}{2}$$

Now according o the problem the two given lines are parallel.

i.e., m$$_{1}$$ = m$$_{2}$$

$$\frac{k}{7}$$ = -$$\frac{1}{2}$$

k = -$$\frac{7}{2}$$

Therefore, the value of k = -$$\frac{7}{2}$$

Equation of a Straight Line