Question 51 markMCQSection A

  1. If a pair of linear equations in two variables is represented by two coincident lines, then the pair of equations has : (a) a unique solution (b) two solutions (c) no solution (d) an infinite number of solutions

Pair of Linear Equations in Two Variables
Consistency - coincident lines

Options

(A)a unique solution
(B)two solutions
(C)no solution
(D)an infinite number of solutions

Official answer

(d) an infinite number of solutions — coincident lines overlap completely, satisfying a1a2=b1b2=c1c2\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}, so every point on the line is a solution.

Answer keywords:coincident linesinfinite solutionsconsistent dependenta1/a2 = b1/b2 = c1/c2pair of linear equationsgraphical methodsame line

Marking scheme

  1. 1.1 mark: correct option (d) an infinite number of solutions.
  2. 2.Expected reasoning: coincident lines share all points; algebraically a1a2=b1b2=c1c2\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}, a consistent (dependent) system with infinitely many solutions.

Hint

Coincident lines lie one on top of the other, so they share every point — think how many common points that gives.

Quick oral answer

Coincident lines lie exactly on top of each other, sharing every point, so the pair of equations has infinitely many solutions; algebraically a1a2=b1b2=c1c2\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}.

Common mistakes

  1. 1.Confusing coincident lines with parallel lines and answering 'no solution' — parallel lines never meet (no solution) but coincident lines overlap completely (infinite solutions).
  2. 2.Choosing 'unique solution', which applies only to intersecting lines that meet at a single point.
  3. 3.Forgetting the ratio condition a1a2=b1b2=c1c2\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} that distinguishes coincident (equal c-ratio) from parallel (unequal c-ratio) lines.

Analysis and explanation

This MCQ tests the graphical and algebraic interpretation of coincident lines for a pair of linear equations.


Concept

  • Coincident lines lie exactly on top of one another, sharing every point.
  • A solution is any point common to both lines, so coincident lines share infinitely many solutions.

Key points

  • Algebraic condition: a1a2=b1b2=c1c2\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} (dependent, consistent system).
  • Intersecting lines (a1a2≠b1b2\frac{a_1}{a_2} \ne \frac{b_1}{b_2}) give a unique solution; parallel lines (a1a2=b1b2≠c1c2\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}) give no solution.

Common mistakes

  • Confusing coincident with parallel lines (parallel → no solution, coincident → infinite solutions).
  • Thinking two lines can have exactly "two" solutions — impossible for straight lines.

Interesting facts

Coincident lines are called a 'dependent and consistent' system because one equation carries no new information — it is just a multiple of the other, like 2x+3y=62x + 3y = 6 and 4x+6y=124x + 6y = 12.

Among all pairs of straight lines there are only three possibilities: exactly one solution (intersecting), no solution (parallel), or infinitely many (coincident) — 'exactly two solutions' can never happen for straight lines.

The ratio test a1a2=b1b2=c1c2\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} lets you decide the number of solutions WITHOUT drawing the graph, which is why it is a favourite one-mark exam tool.

Question details

Q5 of 53 questions

Board
CBSE
Class
Class 10
Subject
Mathematics (Basic)
Year
2026
Section
Section A
Marks
1
Type
MCQ
Chapter
Pair of Linear Equations in Two Variables
Topic
Consistency - coincident lines
Bloom's level
understand
Syllabus topic
Unit 2: Algebra — Pair of Linear Equations in Two Variables: Consistency and Nature of Solutions (Graphical)
Author
CBSE
Reviewed by
Padhantu Academic Team

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Frequently asked questions

How many solutions does a pair of coincident lines have?

Infinitely many. Coincident lines lie exactly on top of one another, so every point on one line is also on the other, giving an infinite number of common points, i.e. infinitely many solutions.

What is the algebraic condition for coincident lines?

For a1x+b1y+c1=0a_1x + b_1y + c_1 = 0 and a2x+b2y+c2=0a_2x + b_2y + c_2 = 0, the lines are coincident when a1a2=b1b2=c1c2\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}. This is a consistent, dependent system with infinitely many solutions.

How are coincident lines different from parallel lines?

Both have a1a2=b1b2\frac{a_1}{a_2} = \frac{b_1}{b_2}, but parallel lines have c1c2\frac{c_1}{c_2} different (no solution, inconsistent), while coincident lines also have c1c2\frac{c_1}{c_2} equal (infinite solutions, consistent-dependent).