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Infinitely many solutions and coincident lines

A pair of linear equations has infinitely many solutions when both equations represent the same line.

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Student-friendly explanation

If one equation is a true multiple of the other, every point on that line satisfies both equations. This is called coincident lines, and it gives infinitely many common solutions. The graph shows one line lying exactly on top of the other.

How to write this in exams

  1. 1

    Start with the exact idea

    A pair of linear equations has infinitely many solutions when both equations represent the same line.

  2. 2

    Then show how to use it

    1. Compare the coefficient ratios. 2. Check whether all three ratios are equal. 3. If yes, conclude the lines are coincident. 4. State infinitely many solutions. 5. If needed, write that every point on the line works.

  3. 3

    Add one concrete example

    The equations 2x + 4y = 8 and x + 2y = 4 represent the same line, so they have infinitely many solutions.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to say there is no solution because the equations look different. Different appearance does not matter if both are the same line.

Definition

A pair of linear equations has infinitely many solutions when both equations represent the same line.

Example

The equations 2x + 4y = 8 and x + 2y = 4 represent the same line, so they have infinitely many solutions.

Rule to remember

For infinitely many solutions, a1/a2 = b1/b2 = c1/c2.

Memory hook

Same line, many answers.

Examples and method

Worked example

For 3x + 6y = 12 and x + 2y = 4, the first equation is three times the second. Both equations represent the same line, so every point on that line is a solution.

Method to apply

1. Compare the coefficient ratios. 2. Check whether all three ratios are equal. 3. If yes, conclude the lines are coincident. 4. State infinitely many solutions. 5. If needed, write that every point on the line works.

Diagram support

Draw one straight line and explain that both equations lie exactly on that same line.

How CBSE asks it

The exam may ask for the number of solutions, the ratio condition, or the graph relation of coincident lines.

Avoid common mistakes

Common confusion

Students may think infinitely many solutions means the answer is uncertain. In fact, it means every point on the same line works.

Common wrong answer

A common wrong answer is to say there is no solution because the equations look different. Different appearance does not matter if both are the same line.

Exam tip

If a1/a2 = b1/b2 = c1/c2, the equations are coincident and the pair has infinitely many solutions.

Quick check

When two linear equations represent the same line, what is the nature of the solution set?

The solution set is infinite. Every point on that line satisfies both equations, so there are infinitely many ordered pairs that work.

Answer writing and exam use

1-mark answer

A pair of linear equations has infinitely many solutions when both equations represent the same line.

2-mark answer

A pair of linear equations has infinitely many solutions when both equations represent the same line. For infinitely many solutions, a1/a2 = b1/b2 = c1/c2. The equations 2x + 4y = 8 and x + 2y = 4 represent the same line, so they have infinitely many solutions.

3-mark answer

If one equation is a true multiple of the other, every point on that line satisfies both equations. This is called coincident lines, and it gives infinitely many common solutions. The graph shows one line lying exactly on top of the other. For infinitely many solutions, a1/a2 = b1/b2 = c1/c2. For 3x + 6y = 12 and x + 2y = 4, the first equation is three times the second. Both equations represent the same line, so every point on that line is a solution. The exam may ask for the number of solutions, the ratio condition, or the graph relation of coincident lines. A common wrong answer is to say there is no solution because the equations look different. Different appearance does not matter if both are the same line.
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