Infinitely many solutions and coincident lines
A pair of linear equations has infinitely many solutions when both equations represent the same line.
In one line: Same line, many answers.
Practice This ConceptUnderstand it
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
Formula and rule
For infinitely many solutions, a1/a2 = b1/b2 = c1/c2.
See it worked out
Follow the example line by line, then use the same method on new questions.
Worked example
- 1For 3x + 6y = 12 and x + 2y = 4, the first equation is three times the second.
- 2Both equations represent the same line, so every point on that line is a solution.
Method you can reuse
- 1Compare the coefficient ratios.
- 2Check whether all three ratios are equal.
- 3If yes, conclude the lines are coincident.
- 4State infinitely many solutions.
- 5If needed, write that every point on the line works.
Sketch it in your answer
Draw one straight line and explain that both equations lie exactly on that same line.
Check yourself
Answer the question in your head before opening the answer.
Try it yourself
When two linear equations represent the same line, what is the nature of the solution set?
Think first, then show answerHide answer
Common wrong answer
A common wrong answer is to say there is no solution because the equations look different. Different appearance does not matter if both are the same line.
Don't mix this up
Exam tip
Write it in the exam
Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.
How CBSE asks it
- 1
Enough for 1 mark
A pair of linear equations has infinitely many solutions when both equations represent the same line.
- 2
Add for 2 marks
Example: The equations 2x + 4y = 8 and x + 2y = 4 represent the same line, so they have infinitely many solutions.
- 3
Add for 3 marks
If one equation is a true multiple of the other, every point on that line satisfies both equations.
Practice this concept with focused MCQs
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