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:

Remember the visual representation:
- Intersecting lines $\rightarrow$ 1 point of contact $\rightarrow$ Unique solution.
- Parallel lines $\rightarrow$ 0 points of contact $\rightarrow$ No solution.
- Coincident lines $\rightarrow$ Infinite points of contact $\rightarrow$ Infinite solutions.

Answer 1

The Correct Option is D

Step 1: What coincident lines mean algebraically.
If two lines $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$ are coincident, the second equation is really just a scalar multiple of the first, so $a_2=ka_1$, $b_2=kb_1$, $c_2=kc_1$ for some constant $k$.
Step 2: Check what happens to a solution of the first equation.
Take any point $(x,y)$ that satisfies $a_1x+b_1y+c_1=0$. Multiplying this whole equation by $k$ gives $ka_1x+kb_1y+kc_1=0$, which is exactly the second equation, so the same point satisfies it too.
Step 3: Conclude how many shared solutions exist.
Since a line already has infinitely many points, and every single one of them satisfies both equations, the pair of coincident lines shares infinitely many common solutions.
\[ \boxed{\text{an infinite number of solutions}} \]