1 markMCQSection A

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

Q: What is the difference between coincident and parallel lines in terms of solutions?

Coincident lines ($a_1/a_2=b_1/b_2=c_1/c_2$) give infinite solutions since they are the same line; parallel but distinct lines ($a_1/a_2=b_1/b_2\ne c_1/c_2$) give no solution since they never meet.

Pair of Linear Equations in Two Variables

Pair of Linear Equations — 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 satisfy $a_1/a_2 = b_1/b_2 = c_1/c_2$ , so every point on the line is a common solution.

coincident linesinfinite solutionspair of linear equationsconsistent systemdependent equationsa1/a2 = b1/b2 = c1/c2

Marking Scheme

11 mark for correctly selecting option (d), infinite number of solutions.

Hint

Coincident lines are literally the same line — every point satisfying one equation also satisfies the other.

Quick Oral Answer

Coincident lines are the same line overlapping perfectly, so every point on it satisfies both equations — giving an infinite number of solutions.

Analysis & Explanation

Tests the NCERT classification of pairs of linear equations by coefficient ratios.

Concept

For $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ : coincident lines occur when $a_1/a_2 = b_1/b_2 = c_1/c_2$ , giving infinitely many common solutions.

Key points

Intersecting lines ( $a_1/a_2 \ne b_1/b_2$ ) → unique solution.
Parallel distinct lines ( $a_1/a_2 = b_1/b_2 \ne c_1/c_2$ ) → no solution.
Coincident lines are the same line drawn twice, so every point satisfies both equations.

Common mistakes

Confusing coincident lines with merely parallel lines and wrongly answering 'no solution'.
Forgetting that a linear pair can never have exactly two solutions (only none, one, or infinite).

Real-world/exam link

This classification is used to quickly judge consistency of equations without solving them fully — useful in word problems on rates and mixtures.

Common Mistakes

1Confusing coincident lines with parallel lines and answering 'no solution'.
2Forgetting the condition $a_1/a_2 = b_1/b_2 = c_1/c_2$ that defines coincident (dependent) lines.
3Mixing up 'unique solution' (intersecting lines) with 'infinite solutions' (coincident lines).

Interesting Facts

The three cases of a pair of linear equations — unique solution, no solution, infinite solutions — correspond exactly to intersecting, parallel, and coincident lines respectively, a cornerstone concept tested almost every year in CBSE Class 10 board exams.

Graphically, coincident lines look identical when plotted, which is why the algebraic ratio test ( $a_1/a_2=b_1/b_2=c_1/c_2$ ) is essential rather than relying only on visual graphs.

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Frequently Asked Questions

How do you identify coincident lines algebraically?

Two lines $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$ are coincident if $a_1/a_2 = b_1/b_2 = c_1/c_2$ , meaning one equation is just a scalar multiple of the other.

What is the difference between coincident and parallel lines in terms of solutions?

Coincident lines ( $a_1/a_2=b_1/b_2=c_1/c_2$ ) give infinite solutions since they are the same line; parallel but distinct lines ( $a_1/a_2=b_1/b_2\ne c_1/c_2$ ) give no solution since they never meet.