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No solution and parallel lines

A pair of linear equations has no solution when the two lines are parallel and never meet.

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Student-friendly explanation

Parallel lines stay at the same distance apart and do not intersect. Since no point is common to both lines, there is no ordered pair that satisfies both equations together. This is an important case in consistency questions and graph-based answers.

How to write this in exams

  1. 1

    Start with the exact idea

    A pair of linear equations has no solution when the two lines are parallel and never meet.

  2. 2

    Then show how to use it

    1. Write both equations in standard form. 2. Compare the ratios a1/a2, b1/b2, and c1/c2. 3. If the first two are equal and the third is different, conclude no solution. 4. On a graph, show parallel lines. 5. State that no common point exists.

  3. 3

    Add one concrete example

    The lines x + y = 3 and x + y = 7 are parallel. They have the same left side pattern but different constants, so they do not meet.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to say the pair has one solution because both equations look similar. Similar form does not guarantee intersection.

Definition

A pair of linear equations has no solution when the two lines are parallel and never meet.

Example

The lines x + y = 3 and x + y = 7 are parallel. They have the same left side pattern but different constants, so they do not meet.

Rule to remember

For no solution, a1/a2 = b1/b2, but a1/a2 is not equal to c1/c2.

Memory hook

Same slope, different intercepts, no meeting point.

Examples and method

Worked example

For 2x + 3y = 6 and 4x + 6y = 15, the ratios of coefficients of x and y are equal, but the constant ratio is different. So the lines are parallel and there is no solution.

Method to apply

1. Write both equations in standard form. 2. Compare the ratios a1/a2, b1/b2, and c1/c2. 3. If the first two are equal and the third is different, conclude no solution. 4. On a graph, show parallel lines. 5. State that no common point exists.

Diagram support

Draw two straight lines with the same direction and equal spacing so that they do not touch anywhere.

How CBSE asks it

The exam may ask for the nature of consistency, the ratio condition, or the graph of a pair with no solution.

Avoid common mistakes

Common confusion

Students often think parallel lines can meet after extension, but in Euclidean geometry they never intersect.

Common wrong answer

A common wrong answer is to say the pair has one solution because both equations look similar. Similar form does not guarantee intersection.

Exam tip

When a1/a2 = b1/b2 but is not equal to c1/c2, the pair has no solution.

Quick check

What does it mean when two linear equations represent parallel lines?

It means the lines never meet, so there is no common solution. The two equations cannot have one ordered pair that satisfies both at the same time.

Answer writing and exam use

1-mark answer

A pair of linear equations has no solution when the two lines are parallel and never meet.

2-mark answer

A pair of linear equations has no solution when the two lines are parallel and never meet. For no solution, a1/a2 = b1/b2, but a1/a2 is not equal to c1/c2. The lines x + y = 3 and x + y = 7 are parallel. They have the same left side pattern but different constants, so they do not meet.

3-mark answer

Parallel lines stay at the same distance apart and do not intersect. Since no point is common to both lines, there is no ordered pair that satisfies both equations together. This is an important case in consistency questions and graph-based answers. For no solution, a1/a2 = b1/b2, but a1/a2 is not equal to c1/c2. For 2x + 3y = 6 and 4x + 6y = 15, the ratios of coefficients of x and y are equal, but the constant ratio is different. So the lines are parallel and there is no solution. The exam may ask for the nature of consistency, the ratio condition, or the graph of a pair with no solution. A common wrong answer is to say the pair has one solution because both equations look similar. Similar form does not guarantee intersection.
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