How to determine if two lines are parallel?

Geometry is one of the branches of maths that deals which points, angles, lines, segments, etc. They help us to determine the spatial relationship between two spaces. The approach for geometry is evident from ancient times to the development of modernized systems in the developing world. Modernized systems are totally reliant on geometry as it is used for designing, constructional works, architecture in choice of material for construction, and many more.

Geometry is a branch of mathematics that deals with the study of shapes and their properties.

The given article is a discussion of parallel lines expressing properties of parallel lines with examples. The article also includes the method of determining a parallel line with the equations of given lines. Along with some sample numerical problems for reference.

What are Parallel Lines?

Parallel lines are defined as the pair of lines that lie at equal distance from each other and intersect each other at the point of infinity. Parallel lines can be extended to infinity in both directions. We can simply understand parallel lines as the lies which are moving along the horizon without intersecting it in the same or opposite direction.

Notebook lines, railway tracks, zebra crossing are some real-life examples of parallel lines which we can observe in our surroundings.

How to determine if two lines are parallel?

Answer:

The two lines are determined to be parallel when the slopes of each line are equal to the others. If the comparison of slopes of two lines is found to be equal the lines are considered to be parallel.

For this, firstly we have to determine the equations of the lines and derive their slopes. If their slopes are found to be equal the lines are proved to be parallel.

For example

Let us consider two lines with equations 6x – 4y = 25 and 9x – 6y = 12 respectively. And, let their slopes be m1 and m2.

Here,

6x – 4y = 25………….(I)

9x – 6y = 12………..(ii)

Now, for m1 solving equation (I) in the form of y = mx + b

=>6x – 4y = 25

=>4y = 6x – 25

Dividing on both sides by 4

=>4y/4 = 6x/4 – 25/4

=>y = 3/2 x – 25/4

=>m1 = 3/2

Again, for m2 solving equation(ii) in the form of y=mx+b

=>9x – 6y = 12

=>6y = 9x – 12

Dividing on both sides by 6

=>6y/6 = 9/6x – 12/6

=>y = 3/2x – 2

=>m2 = 3/2

Hence, the two given lines are parallel as m1 = m2.

Sample Problems

Problem 1. Determine whether the two lines 6x – 4y = 12 and 3x – 2y = 5 are parallel.

Solution:

Let us consider two lines with equations 6x-4y=12 and 3x-2y=5  respectively. And, let their slopes be m1 and m2.

Here,

6x-4y=12………….(I)

3x-2y=5……….(ii)

Now, for m1 solving equation (I) in the form of y=mx+b

=>6x-4y=12

=>4y=6x-12

Dividing on both sides by 4

=>4y/4=6x/4 -12/4

=>y=3/2 x- 12/4

=>m1=3/2

Again, for m2 solving equation(ii) in the form of y=mx+b

=>-3x+2y=5

=>2y=3x+5

Dividing on both sides by 2

=>2y/2=3x/2+5/2

=>m2=3/2

Hence, the two given lines are parallel as m1=m2.

Problem 2. Determine whether the two lines 2/3x+y=5/3  and 2/3x+y=1 are parallel.

Solution:

Let us consider two lines with equations  2/3x+y=5/3 and 2/3x+y=1  respectively. And, let their slopes be m1 and m2.

2/3x+y=5/3…………(I)

2/3x+y=1 …………..(ii)

Now, for m1 comparing  equation (I) with  the form of y=mx+b

=>2/3x+y=5/3

=>y=-2/3x+5/3

=>m1=-2/3

Again, for m2 comparing  equation (ii) with  the form of y=mx+b

=>2/3x+y=1

=>y=-2/3x+1

=>m2=-2/3

Hence, the two given lines are parallel as m1=m2.

Problem 3. Determine whether the two lines 2x-y=-5 and 2x-y=1 are parallel.

Solution:

Let us consider two lines with equations  2x-y=-5   and 2x-y=1  respectively. And, let their slopes be m1 and m2.

2x-y=-5 …………(I)

2x-y=1…………..(ii)

Now, for m1 comparing  equation (I) with  the form of y=mx+b

=>2x-y=5

=>y=2x+5

=>m1=2

Again, for m2 comparing  equation (ii) with  the form of y=mx+b

=>2x-y=1

=>y=2x-1

=>m2=2

Hence, the two given lines are parallel as m1=m2.