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 = -\(\frac{7}{2}\)







10th Grade Math

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