Condition for consistency using ratios
The ratio condition tells whether a pair of linear equations has one solution, no solution, or infinitely many solutions.
Practice This ConceptMain explanation
Teacher explanation
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
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.
Simple analogy
Two equal ratios are not enough; the third ratio decides the final case.
Common confusion
Students often compare only one ratio and ignore the constant term ratio, which leads to a wrong conclusion.
Exam tip
Write all three ratios side by side before deciding the nature of the pair.
Answer writing and exam use
1-mark use
Write the exact meaning of condition for consistency using ratios in one clean line.
2-mark use
Define condition for consistency using ratios and add one example or condition.
3-mark use
Explain condition for consistency using ratios, show the method or example, and mention the common mistake.
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