Coincident Lines
Two lines that lie exactly on top of each other or we can say that one line is exactly lying on top of another line are called coincident lines. Equations of two coincident lines are the same when reduced to the simplest form. For example, x + y = 4 and 2x + 2y = 8 are the equations of two coincident lines. Or we can say the equation of these two lines is x+y=4.
What are Coincident Lines?
Coincident lines are the lines that are on top of each other. They are neither parallel lines nor perpendicular lines but are completely similar. In other words, these are the lines in which one line completely covers the other line in such a way that we can't identify that there are two lines. Both lines appear to be a single line. Look at the given figure of coincident lines. In the figure, L1 and L2 are coincident lines. L1 is on top of L2.
Coincident Lines Equation
The standard form of the equation of a line is y = mx + b, where, m is the slope and b is the y-intercept of the line.
Equation of Coincident Lines: The equation for coincident lines is given by: ax + by = c. When two lines are exactly top on each other, then there could be no other interruption between them. For example, first-line ⇒ 3x +3y = 9 and second-line ⇒ 9x + 9y = 27 are coinciding lines.
In first Line: 3y = 9 - 3x
⇒ y =3 - x
Similarly, x = 3 - y
In second line: 9y = 27 - 9x
⇒ y = (27 - 9x) / 9 = 3 - x
Similarly, x = (27 - 9y) / 9 = 3 - y
From the equation of the first-line and the second-line, we can see that "x" and "y" values are the same. Thus, they are coincident lines.
Coincident Lines Graph
In a graph, coincident lines will look like one line, but there will be two lines on top of each other. Considering the above example, the graphical representation of both coincident lines is given below.
First-line ⇒ 3x +3y = 9 and second-line ⇒ 9x + 9y = 27
When x = 0 , y = 3, When y = 0 , x = 3 in both the lines.
When x = 1 , y = 2, When y = 1 , x = 2 in both the lines.
Plotting these values on a graph, it will look like this:
Coincident Lines Solutions
If two lines are on top of each other then they are considered coincident lines and they have infinite solutions. And the pair of linear equations is consistent. Considering the above example of the coincident lines equation, let's find out the possible solutions. For example, first-line ⇒ 3x +3y = 9 and second-line ⇒ 9x + 9y = 27 are coinciding lines. In first Line: 3y = 9 - 3x
⇒ y =3 - x
⇒ x = 3 - y
In second line: 9y = 27 - 9x
⇒ y = 3 - x
⇒ x = 3 - y
Solutions - when x = 0, y = 3 ; x = 1, y = 2 and so on. When y = 0, x = 3; y = 1, x = 2 and so on. The value of x and y are the same in both lines. Referring above coincident lines graph image, we can see that many solutions are possible on the lines because every point on the lines is common to both the coincident lines. Thus, the x and y values in both equations will be the same, and there are infinite common points and solutions possible.
Difference between Parallel and Coincident Lines
The difference between the two parallel lines and two coincident lines is that parallel lines have the same width or there is a constant space between the parallel lines and the lines never intersect each other whereas coincident lines don't have constant space and they are on top of each other or in other words one line completely covers the other line. Parallel lines do not have common points while coincident lines have infinite common points.
Look at the figure which shows the difference between parallel lines and coincident lines.
Coincident Lines Formula
The formula of coincident lines helps us to check whether the given set of linear equations represent a pair of coincident lines or not. Suppose the pair of linear equations in two variables are given as: \(a_{1}\)x + \(b_{1}\)y + \(c_{1}\) = 0 and \(a_{2}\)x + \(b_{2}\)y + \(c_{2}\) = 0. The lines representing the given equations are called to be coincident lines only if;
\(\dfrac{a_{1}}{a_{2}}\) = \(\dfrac{b_{1}}{b_{2}}\) = \(\dfrac{c_{1}}{c_{2}}\)
This is the coincident lines formula.
Related Articles on Coincident Lines
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FAQs on Coincident Lines
What are Coincident Lines?
Two lines that completely cover each other or we can say lie on top of one another are called coincident lines. They appear as a single line on the graph but in reality, there are two lines on top of each other with infinite common points.
How do you Know if Two Lines are Coincident Lines?
We can find out the two lines are coincident or not by using the formula a1/a2 = b1/b2 = c1/c2. For example, 7x + 14y + 1 = 0 and 14x + 28y + 2 = 0 are the equation of two lines. In equation 1; a1 = 7, b1 = 14, c1 = 1 and in equation 2; a2 = 14, b2 = 28, c2 = 2. Let's apply the formula, a1/a2 = b1/b2 = c1/c2 ⇒ 1/2 = 1/2 = 1/2, thus the given lines are coincident lines.
What is the Formula for Coincident Lines?
If the given two lines are in this form: a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, then the lines representing the given equations can be considered as coincident lines only if; a1/a2 = b1/b2 = c1/c2. So the formula to check whether lines are coincident or not is this, a1/a2 = b1/b2 = c1/c2.
Are Coincident Lines Perpendicular?
Perpendicular lines are the lines intersecting at a 90-degree angle or right angle. So, coincident lines are not perpendicular lines because the perpendicular lines have a common point making a 90-degree angle whereas coincident lines have many common points.
What is the Difference Between Intersecting Lines and Coincident Lines?
When two lines are meeting at a point then we can say that the lines are intersecting each other or intersection of two lines whereas, in coincident lines, the lines exactly overlap each other or one line completely covers the other line and they appear as a single line. Intersecting lines only have one common point whereas coincident lines have many common points as every point that is on one line is also a point on the second line.
What is the Condition for Coincident Lines?
The equation for lines is given in this form ax + by = c. When two lines are coinciding with each other or lie on top of each other, then there is no intercept difference between the lines. For example, first line is 4x + 4y = 8 and second line is 8x + 8y = 16 are coinciding lines. The second line is two times the first line. When simplified, both the equations are actually the same.
What is the Difference Between Coincident Lines and Parallel Lines?
Parallel lines do not share any common points because the lines are parallel at a constant distance whereas two coincident lines have infinite common points and they don't have constant space because lines are on top of each other.