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Unique solution and intersecting lines

A pair of linear equations has a unique solution when the two lines intersect at exactly one point.

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Student-friendly explanation

When two lines cross at one point, that point is the only ordered pair that satisfies both equations. This happens when the equations are consistent and independent. In algebraic form, the coefficients do not make the equations multiples of each other.

How to write this in exams

  1. 1

    Start with the exact idea

    A pair of linear equations has a unique solution when the two lines intersect at exactly one point.

  2. 2

    Then show how to use it

    1. Compare the equations or draw the graph. 2. Check whether the lines intersect once. 3. Solve algebraically if required. 4. Write the ordered pair. 5. Verify both equations.

  3. 3

    Add one concrete example

    The equations x + y = 5 and x - y = 1 intersect at one point, so they have a unique solution.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to call every intersecting pair 'infinite'. That is wrong because only one meeting point means only one solution.

Definition

A pair of linear equations has a unique solution when the two lines intersect at exactly one point.

Example

The equations x + y = 5 and x - y = 1 intersect at one point, so they have a unique solution.

Rule to remember

For one unique solution, the two lines must not be parallel and must not coincide. In ratio form, a1/a2 is not equal to b1/b2.

Memory hook

One crossing point means one answer pair.

Examples and method

Worked example

For x + y = 5 and x - y = 1, adding gives 2x = 6, so x = 3. Substituting gives y = 2. The lines meet at (3, 2), so the solution is unique.

Method to apply

1. Compare the equations or draw the graph. 2. Check whether the lines intersect once. 3. Solve algebraically if required. 4. Write the ordered pair. 5. Verify both equations.

Diagram support

Show two straight lines crossing once, and clearly mark the intersection point with coordinates.

How CBSE asks it

The exam may ask for the nature of the solution, the graph type, or the algebraic condition for a unique solution.

Avoid common mistakes

Common confusion

Students may confuse intersecting lines with parallel lines, especially when the graph is drawn approximately.

Common wrong answer

A common wrong answer is to call every intersecting pair 'infinite'. That is wrong because only one meeting point means only one solution.

Exam tip

If the two lines meet once, write the intersection point as the final answer and check it in both equations.

Quick check

What kind of graph relation gives exactly one solution for a pair of linear equations?

Exactly one solution is obtained when the two lines intersect at one point. That single common point satisfies both equations and gives one ordered pair.

Answer writing and exam use

1-mark answer

A pair of linear equations has a unique solution when the two lines intersect at exactly one point.

2-mark answer

A pair of linear equations has a unique solution when the two lines intersect at exactly one point. For one unique solution, the two lines must not be parallel and must not coincide. In ratio form, a1/a2 is not equal to b1/b2. The equations x + y = 5 and x - y = 1 intersect at one point, so they have a unique solution.

3-mark answer

When two lines cross at one point, that point is the only ordered pair that satisfies both equations. This happens when the equations are consistent and independent. In algebraic form, the coefficients do not make the equations multiples of each other. For one unique solution, the two lines must not be parallel and must not coincide. In ratio form, a1/a2 is not equal to b1/b2. For x + y = 5 and x - y = 1, adding gives 2x = 6, so x = 3. Substituting gives y = 2. The lines meet at (3, 2), so the solution is unique. The exam may ask for the nature of the solution, the graph type, or the algebraic condition for a unique solution. A common wrong answer is to call every intersecting pair 'infinite'. That is wrong because only one meeting point means only one solution.
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