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
From Condition of Parallelism to HOME
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Nov 06, 24 09:18 AM
Nov 05, 24 01:49 PM
Nov 05, 24 09:15 AM
Nov 05, 24 01:15 AM
Nov 05, 24 12:55 AM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.