Question

If two linear equations represent the same line, then the pair of linear equations has _____.

• A. One solution
• B. Two solutions
• C. No solution
• D. Infinitely many solutions

Hint:

Remember: \[ \text{Intersecting} \rightarrow 1 \text{ solution} \] \[ \text{Parallel} \rightarrow 0 \text{ solution} \] \[ \text{Coincident} \rightarrow \infty \text{ solutions} \]

Answer 1

The Correct Option is D

Concept: A pair of linear equations can represent:
• Intersecting lines \(\rightarrow\) one solution
• Parallel lines \(\rightarrow\) no solution
• Coincident lines \(\rightarrow\) infinitely many solutions When two equations represent the same line, the lines overlap completely. Such lines are called coincident lines.

Step 1: Understand what “same line” means. Suppose: \[ x+y=2 \] and: \[ 2x+2y=4 \] The second equation is simply obtained by multiplying the first equation by 2. Both equations represent exactly the same line on the graph.

Step 2: Understand the meaning of solutions. A solution of a pair of linear equations is a point that satisfies both equations simultaneously. Since the two equations are actually the same line:
• Every point on the first line lies on the second line.
• Every point on the second line lies on the first line. Therefore: \[ \text{There are infinitely many common points.} \] Hence: \[ \text{Infinitely many solutions} \]

Step 3: Relation using coefficients. For equations: \[ a_1x+b_1y+c_1=0 \] and: \[ a_2x+b_2y+c_2=0 \] If: \[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \] then the lines are coincident. Hence the system has infinitely many solutions. Final Answer: \[ \boxed{\text{Infinitely many solutions}} \]