Condition for consistency using ratios
The ratio condition tells whether a pair of linear equations has one solution, no solution, or infinitely many solutions.
In one line: Two equal ratios are not enough; the third ratio decides the final case.
Practice This ConceptUnderstand it
For two equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, comparing the ratios of coefficients gives the nature of the solution. This is a fast way to judge consistency without solving fully. It is very useful in theory questions and in checking graph-based answers.
Example
Formula and rule
One solution: a1/a2 not equal to b1/b2.
No solution: a1/a2 = b1/b2 not equal to c1/c2.
Infinitely many: 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 2x + 3y = 12 and 4x + 6y = 18, the ratios of x and y coefficients are equal, but the constant ratio is different.
- 2So the pair has no solution.
Method you can reuse
- 1Convert both equations to standard form.
- 2Find a1/a2, b1/b2, and c1/c2.
- 3Compare them carefully.
- 4State the nature of the solution.
- 5If needed, connect the result to the graph.
Visual tip
A ratio check can be paired with a simple line sketch: intersecting, parallel, or coincident.
Check yourself
Answer the question in your head before opening the answer.
Try it yourself
Why do we compare three ratios in the consistency test for linear equations?
Think first, then show answerHide answer
Common wrong answer
A common wrong answer is to say equal first two ratios always mean infinitely many solutions. The constant term ratio must also match for that conclusion.
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
The ratio condition tells whether a pair of linear equations has one solution, no solution, or infinitely many solutions.
- 2
Add for 2 marks
Example: If a1/a2 is not equal to b1/b2, the pair has one solution. If a1/a2 = b1/b2 but not equal to c1/c2, the pair has no solution.
- 3
Add for 3 marks
For two equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, comparing the ratios of coefficients gives the nature of the solution.
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