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Condition for consistency using ratios

The ratio condition tells whether a pair of linear equations has one solution, no solution, or infinitely many solutions.

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Student-friendly 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.

How to write this in exams

  1. 1

    Start with the exact idea

    The ratio condition tells whether a pair of linear equations has one solution, no solution, or infinitely many solutions.

  2. 2

    Then show how to use it

    1. Convert both equations to standard form. 2. Find a1/a2, b1/b2, and c1/c2. 3. Compare them carefully. 4. State the nature of the solution. 5. If needed, connect the result to the graph.

  3. 3

    Add one concrete 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.

  4. 4

    Avoid this incomplete 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.

Definition

The ratio condition tells whether a pair of linear equations has one solution, no solution, or infinitely many solutions.

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.

Rule to remember

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.

Memory hook

Two equal ratios are not enough; the third ratio decides the final case.

Examples and method

Worked example

For 2x + 3y = 12 and 4x + 6y = 18, the ratios of x and y coefficients are equal, but the constant ratio is different. So the pair has no solution.

Method to apply

1. Convert both equations to standard form. 2. Find a1/a2, b1/b2, and c1/c2. 3. Compare them carefully. 4. State the nature of the solution. 5. If needed, connect the result to the graph.

Diagram support

A ratio check can be paired with a simple line sketch: intersecting, parallel, or coincident.

How CBSE asks it

The exam may ask you to identify consistency from the given equations or to state the meaning of the ratio conditions.

Avoid common mistakes

Common confusion

Students often compare only one ratio and ignore the constant term ratio, which leads to a wrong conclusion.

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.

Exam tip

Write all three ratios side by side before deciding the nature of the pair.

Quick check

Why do we compare three ratios in the consistency test for linear equations?

We compare three ratios to know whether the lines intersect once, never meet, or overlap completely. The relation among all three ratios tells the exact nature of the solutions.

Answer writing and exam use

1-mark answer

The ratio condition tells whether a pair of linear equations has one solution, no solution, or infinitely many solutions.

2-mark answer

The ratio condition tells whether a pair of linear equations has one solution, no solution, or infinitely many solutions. 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. 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-mark answer

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. 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. For 2x + 3y = 12 and 4x + 6y = 18, the ratios of x and y coefficients are equal, but the constant ratio is different. So the pair has no solution. The exam may ask you to identify consistency from the given equations or to state the meaning of the ratio conditions. 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.
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