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:

For equations \(a_{1}x + b_{1}y + c_{1} = 0\) and \(a_{2}x + b_{2}y + c_{2} = 0\), the condition for coincident lines is:
\[ \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \]

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A solution to a system of linear equations corresponds to a point that lies on both lines.
If the lines are coincident, it means one line lies exactly on top of the other, effectively making them the same line.
Step 2: Detailed Explanation:
- If lines intersect at a single point, there is a unique solution.
- If lines are parallel, they never meet, so there is no solution.
- If lines are coincident, every point on one line is also on the other line.
Since a line consists of infinitely many points, there are infinitely many common points.
Step 3: Final Answer:
Therefore, a pair of coincident lines has an infinite number of solutions.