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:

Summary of Linear Equation Solutions: Intersecting lines = Unique solution. Parallel lines = No solution. Coincident lines = Infinite solutions.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A solution to a pair of linear equations corresponds to a point of intersection between their two lines.
Step 2: Key Formula or Approach:
Compare the ratios of coefficients for \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \). For coincident lines: \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \).
Step 3: Detailed Explanation:
1. Coincident lines lie exactly on top of each other. 2. This means every single point on the first line is also a point on the second line. 3. Since a line consists of an infinite number of points, there are infinitely many points that satisfy both equations simultaneously.
Step 4: Final Answer:
The pair of equations has an infinite number of solutions.