Infinitely many solutions and coincident lines
A pair of linear equations has infinitely many solutions when both equations represent the same line.
Practice This ConceptMain explanation
Teacher explanation
If one equation is a true multiple of the other, every point on that line satisfies both equations. This is called coincident lines, and it gives infinitely many common solutions. The graph shows one line lying exactly on top of the other.
Example
The equations 2x + 4y = 8 and x + 2y = 4 represent the same line, so they have infinitely many solutions.
Simple analogy
Same line, many answers.
Common confusion
Students may think infinitely many solutions means the answer is uncertain. In fact, it means every point on the same line works.
Exam tip
If a1/a2 = b1/b2 = c1/c2, the equations are coincident and the pair has infinitely many solutions.
Answer writing and exam use
1-mark use
Write the exact meaning of infinitely many solutions and coincident lines in one clean line.
2-mark use
Define infinitely many solutions and coincident lines and add one example or condition.
3-mark use
Explain infinitely many solutions and coincident lines, show the method or example, and mention the common mistake.
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Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.
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