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

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.