Coincident Lines: Meaning, Properties, Formula & Examples

Coincident Lines

Coincident lines are two or more lines lying precisely on top of one another and sharing all their points, basically creating the same line. In contrast to parallel lines that never meet, coincident lines overlap entirely, and their equations are multiples of or equal to each other. In geometry, the coincident lines meaning refers to lines that lie exactly on top of each other, sharing every point in common.

For instance, the lines given by Coincident lines equation x+y=4x+y=4 and 2x+2y=82x+2y=8 are coincident since they both lie on the same line. The concept of coincident lines is beneficial in solving systems of equations and studying geometric configurations.

Read more: Cartesian Plane

What Are Coincident Lines?

Coincident Lines meaning - Coincident lines are two or more lines perfectly on top of one another, sharing each point. This indicates they share the same slope and the same y-intercept. Even if they may seem like different lines in equation form, they actually are the same line in the coordinate plane.

To understand what are coincident lines, remember that they have the same slope and intercept, which makes them overlap completely on a graph.

For example, the Coincident lines equation y=2x+1 and 2y=4x+2 are coincident lines. If you reduce the second equation, it becomes the same as the first.

Read More: 30-60-90 Triangle 

Coincident Lines Equation

The standard coincident lines equation is y = mx + b,

Where

• m is the slope

• b is the y-intercept of the line

Coincident lines are two or more linear equations that are scalar multiples of one another. That is, they have the same slope and the same y-intercept.

Example 1: Basic Equations

Look at the following two equations:

y=2x+1

2y=4x+2

If you divide each of the terms of the second equation by 2, you get y=2x+1. This indicates that both equations express the same line and are therefore coincident.

Example 2: More Complicated Equations

The following two equations are also coincident:

3x+6y=12

x+2y=4

To see if they are coincident, you can solve each of them to determine their slope and y-intercept.

For the first equation:

6y=−3x+12

y=−63​x+612​

y=−21​x+2

For the second equation:

2y=−x+4

y=−21​x+2

Because both of these equations reduce to the same form (y=−21​x+2), they are coincident lines.

Enroll Now for Maths Online Classes

Coincident Lines Graph

Coincident lines graph as a single, solid line on a graph because they take up the same region of space in the coordinate plane. While they are equated by two or more different equations, all of the equations reduce to the same form, which gives them the same slope and y-intercept.

For instance

Look at the following two equations that are coincident lines:

y=2x+1

2y=4x+2

If you reduce the second equation by dividing all terms by 2, then you have y=2x+1, which is the same as the first equation. This is why they're coincident and become one line on the graph below.

Read More: Before Number Concept

Coincident Lines Solutions

Coincident lines have infinitely many solutions since they are, in essence, the same line. Because each point on one line is a point on the other, they intersect at every point from start to finish along their length.

Example 1: Basic Equations

Suppose we have two equations:

y=3x+2

2y=6x+4

To check that they are on the same line, you can plug in some points.

If x=1, the first equation produces y=3(1) + 2 = 5.

For the second equation, 2y=6(1)+4, which simplifies to 2y=10, so y=5.

The two equations have the point (1, 5) in common. This is a point for any point you pick, validating an infinite number of solutions.

Example 2: Non-Simplified Form

Consider these two equations:

4x−2y=8

2x−y=4

If you multiply the second equation by 2, it becomes 2(2x−y)=2(4), which is 4x−2y=8.

As the two equations become the same after this operation, they are coincident lines, and each point that satisfies one equation also satisfies the other.

Coincident Lines Formula

The coincident lines formula assists us in verifying whether the given pair of lines is a pair of coincident lines or not.

Take, for instance, two linear equations:

a1x+b1y+c1=0

a2x+b2y+c2=0

These lines are coincident if the ratios of the corresponding coefficients are equal:

a1/a2=b1/b2=c1/c2

For example, the line

3x+3y=9

9x+9y=27

These are coincident since the second is a scalar multiple of the first. This implies that the two lines are perfectly on top of one another and share infinitely many points.

Another Coincident lines example:

The lines 9x−2y+16=0 and 18x−4y+32=0 are coincident since

9/18=−2/−4=16/32

All ratios are equal, affirming coincidence.

Read More: Mean, Median, Mode

Coincident Lines Examples

Coincident lines are two or more linear equations that represent the same line. They have identical slopes and y-intercepts, and one equation is a scalar multiple of the other. Below are some coincident lines examples. 

Example 1: Simple Multiplication

Line 1: y=2x+3

Line 2: 2y=4x+6

Explanation: Dividing the second equation by 2 gives y=2x+3, which is identical to the first. They are coincident.

Example 2: Standard Form

Line 1: 3x−5y=15

Line 2: 6x−10y=30

Explanation: The second equation is just the first one multiplied by 2. We can check this using the ratio of coefficients:

6/3​=−10/−5​=30/15​=2/1​.

All ratios are equal, so the lines are coincident.

Example 3: Rearranging Terms

Line 1: y=−x+2

Line 2: 3y+3x=6

Explanation: If we divide the second equation by 3, we obtain y+x=2. Rearranging to solve for y, we get y=−x+2, which is identical to the first equation.

Example 4: Negative Multipliers

Line 1: y=−2x+4

Line 2: −4y=8x−16

Explanation: If you divide the second equation by -4, you obtain y=(−8/4​x)−16/-4)​, or y=−2x+4. This is the same as the first equation.

Example 5: Fractional Coefficients

Line 1: y=1​x/2+1

Line 2: 4y=2x+4

Explanation: The second equation divided by 4 is y=2​x/4+4/4​, or y=1​x/2+1. Both equations graph the same line.

Read More: Construction in Maths

Difference Between Coincident and Parallel Lines

Coincident lines are lines that exactly sit on top of one another and have all their points in common, constituting basically the same line. Parallel lines, on the other hand, are always equidistant from one another and never cut each other. Below are the main differences between coincident and parallel lines:

How CuriousJr Makes Learning Mathematics Simple for Your Child?

CuriousJr online Mental Maths classes makes math learning easy by turning it into an engaging, gamified journey. With fun challenges, quizzes, and interactive classes, the online coaching transforms complex concepts into simpler ones. There are also personalized learning paths, ensuring that every child can build a strong mathematical foundation at their own pace. Book a demo class today to know more about our classes.