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 xaxis. 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 = \(\frac{7}{2}\)
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