parallel lines

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Parallel Lines

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Parallel Lines

Two lines with the same slope are said to be a parallel lines. When we plot a graph of parallel lines then they will never intersect to each other.

Line AB and Line CD are Parallel lines and it is denoted by AB||CD.

The equation of the straight lines are y = m1x + c and y = m2x + c are parallel if the slopes of both lines are equal i.e.

m1 = m2

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A B

C D



Parallel Lines

Q. Are the lines 12x + 3y = – 9 and – 8x – 2y = 14 are parallel?

Solution:

First find the slope of both the lines

12x + 3y = – 9 –8x – 2y = 14

3y = –12x – 9 –2y = 8x + 14

y = –4x – 3 y = –4x – 7

Since the slopes of both the lines are equal, so the lines are parallel.


Slope = –4 Slope = –4

y = –4x – 3

y = –4x – 7



Parallel Lines

Practice Question :

Are the given lines are parallel?

1. 6x – 3y = 5 and 2y = 4x – 4

2. x + 2y = 6 and 2x – 2y = 6

3. 5x – y = 5 and y = 10x – 4

4. 3x – 3y = 6 and x – y = 6



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Constructing Parallel Lines

The online version of this tutorial can be accessed at www.freematheducation.com . It provides all the basics and the concept in details including more practice questions with complete step by step solutions.


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We can find the equation of a line parallel to the a given line which is passing through the point by

finding the slope m of the given line;

Finding the equation of a line through the given point (x1, y1) with the slope equals to m.

The slope intercept form of a line is y = mx + c, where m is the slope of the line and c be its y intercept.

The equation of a line passing through point (x1, y1) with slope m is given by

y – y1 = m (x – x1)





Q. Find the equation of a line which is parallel to the line 2x + 6y = 12 and passing through point (2, –1).

Solution:

To find the equation of a line, first find the slope of given line

6x + 2y = 12

2y = – 6x + 12

y = – 3x + 6

The slope of a line is -3. Since the slopes of parallel lines are equal. Therefore the equation of a line is given by

y – y1 = m (x – x1)

y – (–1) = –3 (x – 2)

y + 1= –3x + 6

3x + y = 5www.freematheducation.com 7

Slope = –3

Compare with

Slope Intercept Form

y = mx + c




Practice Question :

1. Find the equation of the line going through the point (4, –1) and parallel to x + 2y = 10

2. Find the equation of the line going through the point (–2, –7) and parallel to 2x – 3y = 12



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Perpendicular Lines



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Perpendicular Lines

Perpendicular lines are the lines that intersect in a right angle.

We can find the equation of a line perpendicular to a given line and going through a given point by:

finding the slope m of the given line;

Finding the equation of a line through the given point (x1, y1) with the slope equals to -1/m.


The equation of a line passing through point (x1, y1) with slope m is given by

y – y1 = m (x – x1)

The equation of the straight lines are y = m1x + c and y = m2x + c are perpendicular the slopes are negative reciprocal to each other i.e. m1m2 = -1.




Perpendicular Lines

Q. Are the lines 2x + y = 5 and x – 2y = – 4 are perpendicular?

Solution:

First find the slope of both the lines

2x + y = 5 x – 2y = –4

y = –2x + 5 –2y = –x – 4

y = ½ x +2

Since the slopes of both the lines are negative reciprocal to each other, so both

the lines are perpendicular lines.


Slope m1 = –2

Slope m2 = ½

2x + y = 5

x – 2y = – 4

m1 m2 = –2 x ½ = –1



Perpendicular Lines

Practice Question :

Are the given lines are parallel?

1. 6x – 3y = 5 and 3x + 6y = 14

2. 3x – 3y = 6 and 3x + 3y = 8

3. x + 2y = 6 and x – 2y = 6

4. y = 10x – 12 and y = 10x – 4



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Constructing Perpendicular Lines



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We can find the equation of a line perpendicular to the given line which is passing through the point by

finding the slope m1 of the given line;

Find the negative reciprocal of that slope i.e. m2 = 1/m1

Finding the equation of a line through the given point (x1, y1) with the slope equals to m2.


The equation of a line passing through point (x1, y1) with slope m2 is given by

y – y1 = m2 (x – x1)





Q. Find the equation of a line given through the point (3, 0) and perpendicular to 6x + 2y = 12.

Solution:

To find the equation of a line, first find the slope of given line

6x + 2y = 12

2y = – 6x + 12

y = – 3x + 6

Slope of perpendicular line is m2 = –(1/ –3) = 1/3. Therefore the equation of a line perpendicular to given line is

y – y1 = m2 (x – x1)

y – (0) = 1/3 (x – 3)

y = –1/3 x – 1

x + 3y + 3 = 0


Slope = –3

Compare with

Slope Intercept Form

y = mx + c




Practice Question :

1. Find the equation of the line going through the point (4, –1) and perpendicular to x + 2y = 10.

2. Find the equation of the line going through the point (–2, –7) and perpendicular to 2x – 3y = 12



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Answer Key



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Answer Key

Your practice Session is completed. Your Answer Key is

Parallel Lines :

1. Yes 2. No 3. No 4. Yes

Constructing Parallel Lines :

1. x – y = 2 2. 2x – 3y = 17

Perpendicular Lines :

1. Yes 2. Yes 3. No 4. No

Constructing Perpendicular Lines :

1. 2x – y = 9 2. 3x + 2y = – 20




Parallel Lines


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