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:

A quick check of the ratios of coefficients:
If $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, then the equations are dependent and equivalent, which graphically translates to coincident lines and algebraically to infinitely many solutions.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This question relates to the system of linear equations in two variables and their graphical representation.
We need to determine the nature of the solutions when the two lines representing the equations coincide on the coordinate plane.

Step 2: Key Formula or Approach:
Let the pair of linear equations be:
\[ a_1x + b_1y + c_1 = 0 \]
\[ a_2x + b_2y + c_2 = 0 \]
The geometrical and algebraic behavior of these lines is determined by their coefficient ratios:

• Intersecting lines: $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$ (Consistent system with a Unique solution)

• Parallel lines: $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$ (Inconsistent system with No solution)

• Coincident lines: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ (Dependent consistent system with Infinitely many solutions)


Step 3: Detailed Explanation:

• Coincident lines are lines that lie directly on top of each other.

• This means that any point $(x, y)$ that lies on the first line also lies on the second line.

• Since a straight line is made up of an infinite number of points, the two coincident lines share all of their points.

• A solution to a system of linear equations is defined as a point of intersection of the lines representing them.

• Since the lines intersect at every single point along their length, there are infinitely many points of intersection.

• Consequently, the pair of linear equations has an infinite number of solutions.


Step 4: Final Answer:
Coincident lines have an infinite number of solutions, which corresponds to option (D).