## Introduction to Coincident Lines

Coincident lines are the lines that coincide or lie on top of each other. So far we have learned about different types of lines in Geometry, such as parallel lines, perpendicular lines, with respect to a two-dimensional or three-dimensional plane. In the case of parallel lines, they are parallel to each other and they are at a defined distance between them. On the other hand, perpendicular lines are lines that intersect each other at 90 degrees. Both parallel lines and perpendicular lines do not coincide with each other. Here comes the concept of coincident lines when both the lines coincide with each other. In this article, we will learn about the coincident lines equation, coincident lines condition along with coincident lines solution.

In the below diagram we have represented parallel lines, intersecting lines and coincident lines.

In the case of parallel lines, there are no common points. Intersecting lines have one point in common and coincident lines have infinitely many points in common.

Apart from these three lines, there are many different lines that are neither parallel, perpendicular, nor coinciding lines. These lines can be oblique lines or intersecting lines, which intersect at different angles, instead of perpendicular to each other.

## Coincident Lines Definition

The word ‘coincide’ means that it happens at the same time. In Mathematics, the coincident is defined as the lines that lie upon each other. It is placed in such a way that when we look at them, they appear to be a single line, instead of double or multiple lines.

If we see the given below diagram of coincident lines, it appears as a single line, but in actual we have drawn two different lines here. First, we drew a line of purple colour and then on top of it drew another line which is of black colour.

### Coincident Lines Equation

We have learned that linear equations can be represented by the equation y=mx+c, where m and c are real numbers.

If we consider the equation of a line, the standard form is:

y = mx + b

Where m is the slope of the line and b is the intercept made by the line.

Equation of Parallel Lines:

Now, in the case of two lines that are parallel to each other, we represent the equations of the lines as:

y = m_{1}x + b_{1}

And y = m_{2}x + b_{2}

For example, y = x + 2 and y = 2x + 4 are parallel lines. Here, the slope of both the lines is equal to 2 and the intercept difference between them is 2. Hence, they are parallel lines at a distance of 2 units.

Equation of Coincident Lines:

The equation for lines is given below;

ax + by = c

When two lines are coinciding with each other, then there is no intercept difference between them.

For example, consider the equation of two coinciding lines,

x + y = 4 and 2x + 2y = 8 are two coinciding lines.

If we compare both the lines we get the second line is twice the first line.

If we put ‘y’ on the Left-hand side and the rest of the equation on the Right-hand side, then we get the following result;

First-line y = 2 - x ..(i)

Second-line 2y = 4 - 2x

2y = 2(2 - x)

y = 2 - x …(ii)

From equation (1) and (2), we can conclude both the lines are the same.

Hence, they coincide with each other.

### Coincident Lines Formula

Consider a_{1}x + b_{1}y + c_{1 }= 0 and a_{2}x + b_{2}y + c_{2 }= 0 be the pair of linear equations in two variables. The lines representing these equations are said to be coincident if follow the below condition

Given below is the Coincident lines condition

\[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\]

Here, the given pair of equations is called consistent equations and they can have infinitely many solutions.

### Coincident Lines Solutions

Refer to the given below solved example to understand how to use the formula of coincident lines.

Check whether the lines representing the pair of equations 9x-2y+16=0 and 18x-4y+32=0 are coincident or not.

Ans: Given equation

9x - 2y + 16 = 0

18x - 4y + 32 = 0

We will compare the above equation with a_{1}x + b_{1}y + c_{1}=0 and a_{2}x + b_{2}y +c_{2}=0

After comparing we get a_{1}= 9 b_{1 }= -2 and c_{1}= 16

a_{2 }= 18 b_{2 }= -4 and c_{2} = 32

Now, calculate the value of

\[ \frac{a_{1}}{a_{2}}=\frac{9}{18}=\frac{1}{2}\]

\[ \frac{b_{1}}{b_{2}}=\frac{-2}{-4}=\frac{1}{2}\]

\[\frac{c_{1}}{c_{2}}=\frac{16}{32}=\frac{1}{2}\]

Hence from above, we can conclude that

\[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\]

Therefore, the lines representing the above equations are coincident.

This can be shown graphically as:

1. Do Coincidence Lines are the Same as Parallel Lines?

Ans: There is a slight difference between the two parallel lines and two coincident lines. Parallel lines have constant space between them while coincident don't. Parallel lines do not have points in common while coincident lines have all points in common.

2. How Can We Represent the Coincident Lines?

Ans: Suppose the given equation of a line as ax + by = c, here y-intercept is equal to c/b. If each line in the system has the same slope as well as the same y-intercept then the lines are coincident.

3. What Does a Coincident Line Look Like?

Ans: Two lines or shapes that lie exactly on top of each other are said to be coincident. If we look at the coincident line, it will look like a single line, because they are on top of each other we can't see both the lines.