Question

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

Hint:

Remember the visual representation:
- Intersecting lines $\rightarrow$ 1 point of contact $\rightarrow$ Unique solution.
- Parallel lines $\rightarrow$ 0 points of contact $\rightarrow$ No solution.
- Coincident lines $\rightarrow$ Infinite points of contact $\rightarrow$ Infinite solutions.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks about the nature of the solutions for a system of linear equations in two variables when their graphical representations are coincident lines.

Step 2: Key Formula or Approach:
Let the system of linear equations in two variables be:
\[ a_1x + b_1y + c_1 = 0 \]
\[ a_2x + b_2y + c_2 = 0 \]
The relationship between the coefficients and the nature of the lines is given by:
1. Intersecting lines: $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$ (Consistent system with a unique solution).
2. Coincident lines: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ (Dependent consistent system with infinitely many solutions).
3. Parallel lines: $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$ (Inconsistent system with no solution).

Step 3: Detailed Explanation:

• A solution to a system of two linear equations in two variables corresponds to a point of intersection of their graphical lines.

• Coincident lines are lines that lie directly on top of each other. Mathematically, they represent the same line, just expressed in different algebraic forms.

• Since a straight line contains an infinite number of points, two coincident lines share all of their points in common.

• Every single coordinate pair $(x, y)$ that satisfies the first equation will also satisfy the second equation.

• Since there are infinitely many such coordinate pairs on any line, the system has an infinite number of common solutions.


Step 4: Final Answer:
Coincident lines represent a pair of equations with infinitely many solutions.
Hence, option (D) is correct.