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 algebraic condition for coincident lines?

For $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$, the lines are coincident when $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$. This is a consistent, dependent system with infinitely many solutions.

Q: How are coincident lines different from parallel lines?

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

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 $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ , so every point on the line is a solution.

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

Marking Scheme

11 mark: correct option (d) an infinite number of solutions.
2Expected reasoning: coincident lines share all points; algebraically $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ , 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 $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ .

Analysis & 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: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ (dependent, consistent system).
Intersecting lines ( $\frac{a_1}{a_2} \ne \frac{b_1}{b_2}$ ) give a unique solution; parallel lines ( $\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}$ ) 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.

Common Mistakes

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

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 $2x + 3y = 6$ and $4x + 6y = 12$ .

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

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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 $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ , the lines are coincident when $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ . This is a consistent, dependent system with infinitely many solutions.

How are coincident lines different from parallel lines?

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