Coincident

Coincident

Identical, one superimposed on the other. That is, two or more geometric figures that share all points. For example, two coincident lines would look like one line since one is on top of the other.

See also

Concurrent

Worked Example

Problem: Determine whether the lines given by the equations 2x + 3y = 6 and 4x + 6y = 12 are coincident, parallel, or intersecting.

Step 1: Write both equations in slope-intercept form. For the first equation, solve for y:

2x + 3y = 6 \implies y = -\tfrac{2}{3}x + 2

Step 2: Solve the second equation for y:

4x + 6y = 12 \implies 6y = -4x + 12 \implies y = -\tfrac{2}{3}x + 2

Step 3: Compare the two results. Both lines have slope

m = -\tfrac{2}{3}

and y-intercept

b = 2

. The equations describe exactly the same line.

Step 4: Check by observing that the second equation is simply 2 times the first equation:

2(2x + 3y) = 2(6) \implies 4x + 6y = 12

Answer: The two lines are coincident. They share every point, so the system has infinitely many solutions.

Another Example

Problem: Are the lines x - y = 1 and 3x - 3y = 5 coincident?

Step 1: Divide the second equation by 3 to simplify:

3x - 3y = 5 \implies x - y = \tfrac{5}{3}

Step 2: Compare with the first equation

x - y = 1

. The left sides are identical, but

1 \neq \tfrac{5}{3}

. The lines have the same slope but different y-intercepts.

Answer: The lines are parallel, not coincident. They never intersect and do not share any points.

Frequently Asked Questions

How many solutions does a system of coincident lines have?

A system of two coincident lines has infinitely many solutions. Every point on one line is also on the other, so there is no single unique intersection — the entire line is the solution set.

What is the difference between coincident lines and parallel lines?

Coincident lines have the same slope and the same y-intercept, so they overlap completely and share all points. Parallel lines have the same slope but different y-intercepts, so they never meet and share no points at all.

Coincident lines vs. Concurrent lines

Coincident lines lie exactly on top of each other, sharing every point. Concurrent lines are distinct lines that all pass through a single common point. Two coincident lines overlap entirely; two concurrent lines only share one point.

Why It Matters

Recognizing coincident lines is essential when solving systems of linear equations. If the two equations represent coincident lines, the system is consistent but dependent, meaning it has infinitely many solutions rather than one unique solution or none. This concept also appears in geometry when determining whether shapes, segments, or circles occupy the same position.

Common Mistakes

Mistake: Confusing coincident lines with parallel lines because both have the same slope.

Correction: Same slope alone only guarantees the lines are parallel. To be coincident, the lines must also have the same y-intercept (or, equivalently, one equation must be a scalar multiple of the other). Always check both slope and intercept.

Mistake: Saying a system of coincident lines has "no solution" because there is no single intersection point.

Correction: Coincident lines share every point, so the system actually has infinitely many solutions. A system with no solution arises from parallel (non-coincident) lines.

Related Terms

Line — Fundamental figure that can be coincident
Point — Shared by all coincident figures
Concurrent — Lines meeting at one point, not overlapping
Parallel Lines — Same slope but different intercepts
Geometric Figure — General category of shapes that can coincide
Systems of Equations — Coincident lines give infinitely many solutions