Coincident Lines: Definition, Condition & Examples
Coincident Lines: Definition, Condition & Examples
TL;DR
Coincident lines are two lines that lie exactly on top of each other, sharing every point and therefore infinitely many solutions. This guide defines coincident lines, gives the coefficient-ratio condition ( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} ), separates them from parallel and intersecting lines, and works through six examples.
What Are Coincident Lines?
Coincident lines are two straight lines that occupy the same position in a plane. One sits perfectly on the other, so every point of one line is also a point of the other. Where two ordinary lines might cross at a single point, coincident lines agree at every point. They are, geometrically, a single line written in two different ways.
Because they share all their points, a system of two coincident-line equations has infinitely many solutions: every point on the line satisfies both equations at once. This is one of the three outcomes you can get from a pair of straight-line equations, alongside one solution (lines cross) and no solution (lines are parallel).
The key idea to hold: coincident lines are the same line in disguise — one equation is just a scaled copy of the other.
The Condition For Coincident Lines
Write two lines in standard form:
[ a_1 x + b_1 y + c_1 = 0 ] [ a_2 x + b_2 y + c_2 = 0 ]
The two lines are coincident when all three coefficient ratios are equal:
[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} ]
This says the second equation is just the first multiplied through by a constant. Compare it with the neighbouring cases so the boundaries are clear:
| Relationship | Condition | Number of solutions |
|---|---|---|
| Intersecting | ( \frac{a_1}{a_2} \ne \frac{b_1}{b_2} ) | Exactly one |
| Parallel | ( \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2} ) | None |
| Coincident | ( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} ) | Infinitely many |
The single difference between parallel and coincident is the third ratio. Parallel lines match on the first two ratios (same direction) but disagree on the constant (different position). Coincident lines match on all three. If even the first two ratios differ, the lines simply cross — they are ordinary intersecting lines.
There is also a slope reading of the same fact: coincident lines have the same slope and the same y-intercept, while parallel lines share the slope but have different intercepts.
Examples of Coincident Lines
These run from spotting coincident lines by inspection to checking the ratio condition and reading a real situation. Each problem statement is bold; the steps are plain.
Example 1
Are the lines ( x + y = 4 ) and ( 2x + 2y = 8 ) coincident?
Divide the second equation through by 2:
[ \frac{2x + 2y}{2} = \frac{8}{2} ]
[ x + y = 4 ]
The reduced second equation is identical to the first.
Final answer: yes, the lines are coincident.
Example 2
Are the lines ( 3x + 6y = 9 ) and ( x + 2y = 5 ) coincident?
Check the ratios:
[ \frac{a_1}{a_2} = \frac{3}{1} = 3, \quad \frac{b_1}{b_2} = \frac{6}{2} = 3, \quad \frac{c_1}{c_2} = \frac{9}{5} = 1.8 ]
The first two ratios match, but the third does not. That breaks the coincident condition.
Final answer: no, the lines are parallel, not coincident.
Example 3
For what value of ( k ) are ( 2x + 3y = 7 ) and ( 4x + 6y = k ) coincident?
For coincidence, all three ratios must match. The first two already do:
[ \frac{2}{4} = \frac{3}{6} = \frac{1}{2} ]
So the constant ratio must also equal ( \frac{1}{2} ):
[ \frac{7}{k} = \frac{1}{2} ]
[ k = 14 ]
Final answer: ( k = 14 ).
Example 4
Write a second equation that is coincident with ( 5x - y = 3 ).
Multiply through by 3:
[ 15x - 3y = 9 ]
Final answer: ( 15x - 3y = 9 ).
Example 5
A system is ( y = 2x + 1 ) and ( 2y = 4x + 2 ). How many solutions does it have?
Reduce the second equation by dividing by 2:
[ y = 2x + 1 ]
It matches the first exactly, so the two equations describe one line.
Final answer: infinitely many solutions.
Example 6
A shop prices apples at "2 for ₹40." A poster also reads "₹20 each." Are these the same line?
Let ( x ) be quantity and ( y ) be cost. "2 for ₹40" gives ( y = 20x ). "₹20 each" is also ( y = 20x ).
Final answer: yes, they are coincident lines.
Why Coincident Lines Matter: "When Two Equations Say One Thing"
Coincident lines are how mathematics flags redundant information. If a system of equations turns out to be coincident, the second equation told you nothing the first did not.
- Solving systems. Before grinding through elimination, checking the coefficient ratios tells you instantly whether to expect one answer, none, or infinitely many.
- Data and modelling. Two measurements that produce coincident equations are duplicate readings.
- Calibration. When two sensors are perfectly aligned, their response lines coincide.
Common Mistakes With Coincident Lines
Mistake 1: Checking only two of the three ratios
Where it slips in: Seeing the ratios match and declaring the lines coincident without checking the constant.
Mistake 2: Confusing coincident with parallel
Where it slips in: Treating "same slope" as enough to call lines coincident.
Mistake 3: Expecting one neat solution from a coincident system
Where it slips in: Running elimination on a coincident system and panicking when every variable cancels.
Conclusion
- Coincident lines lie exactly on top of each other and share every point.
- A coincident system has infinitely many solutions; in standard form, ( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} ).
- They share the same slope and intercept; parallel lines share only the slope.
- In a system, an all-cancel result of 0=0 is the fingerprint of coincident equations.