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

• Inclination of a Line
• Slope of a Line
• Intercepts Made by a Straight Line on Axes
• Slope of the Line Joining Two Points
• Equation of a Straight Line
• Point-slope Form of a Line
• Two-point Form of a Line
• Equally Inclined Lines
• Slope and Y-intercept of a Line
• Condition of Perpendicularity of Two Straight Lines
• Condition of parallelism
• Problems on Condition of Perpendicularity
• Worksheet on Slope and Intercepts
• Worksheet on Slope Intercept Form
• Worksheet on Two-point Form
• Worksheet on Point-slope Form
• Worksheet on Collinearity of 3 Points
• Worksheet on Equation of a Straight Line

10th Grade Math

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