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medium difficulty6 min

No solution and parallel lines

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

In one line: Same slope, different intercepts, no meeting point.

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Understand it

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.

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.

Formula and rule

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

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

  1. 1For 2x + 3y = 6 and 4x + 6y = 15, the ratios of coefficients of x and y are equal, but the constant ratio is different.
  2. 2So the lines are parallel and there is no solution.

Method you can reuse

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

Sketch it in your answer

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer
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.

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.

Don't mix this up

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

Exam tip

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

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

The exam may ask for the nature of consistency, the ratio condition, or the graph of a pair with no solution.
  1. 1

    Enough for 1 mark

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

  2. 2

    Add for 2 marks

    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.

  3. 3

    Add for 3 marks

    Parallel lines stay at the same distance apart and do not intersect.

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