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

To solve this problem, we must understand when a pair of linear equations in two variables is represented by coincident lines.

1. Consider the general form of a pair of linear equations in two variables:
   • Equation 1: \(a_1x + b_1y + c_1 = 0\)
   • Equation 2: \(a_2x + b_2y + c_2 = 0\)
2. For these lines to be coincident (exactly the same), the coefficients must be proportional. Specifically, they must satisfy:
   • \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\)
3. When two lines are coincident, every point on one line is also on the other line, resulting in an infinite number of solutions.
4. Let's analyze the given multiple-choice options:
   • Unique solution: Occurs when two lines intersect at exactly one point. This is not the case for coincident lines.
   • Two solutions: Two solutions suggest there are only two intersection points, which is not possible for linear equations.
   • No solution: Occurs when lines are parallel but not coincident. Thus, they never intersect.
   • Infinite number of solutions: Correct option, as coincident lines are the same line, thereby having infinite solutions.
5. Based on the above explanation, the correct choice is: an infinite number of solutions.