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

The question asks about the nature of solutions for a pair of linear equations represented by coincident lines. Let's understand and solve this step-by-step:

When two lines are coincident, they lie exactly on top of each other. Geometrically, this means that every point on one line is also on the other line, hence they are the same line.

Mathematically, if two lines are coincident, their equations can be expressed in such a way that they are proportionate to one another:

• Consider two linear equations: \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\).
• For these two lines to be coincident, the condition \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\) must hold true.

Since coincident lines represent equations that are essentially the same, every point on one line solves the equation of the other. Therefore, they result in an infinite number of solutions.

Let's rule out the incorrect options for clarity:

• Unique Solution: This occurs when lines intersect at exactly one point, which is not the case here.
• Two Solutions: A linear equation pair cannot have exactly two solutions.
• No Solution: This occurs when lines are parallel but distinct, not coincident.

Thus, the correct answer is that the pair of equations has an infinite number of solutions.