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Intersection of Two Linear Polynomials

The intersection of two linear graphs is the point where the two lines meet. Algebraically, it represents the common solution of the two linear equations.

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Student-friendly explanation

When two straight lines meet at one point, the coordinates of that point satisfy both equations. Solving the pair of equations gives the same ordered pair as the intersection point on the graph. This connects algebraic solving with graphical meaning.

How to write this in exams

  1. 1

    Start with the exact idea

    The intersection of two linear graphs is the point where the two lines meet. Algebraically, it represents the common solution of the two linear equations.

  2. 2

    Then show how to use it

    Draw or solve both equations, find the common ordered pair, check it in both equations, and state it as the intersection point.

  3. 3

    Add one concrete example

    The lines x + y = 6 and x - y = 2 intersect at (4, 2) because 4 + 2 = 6 and 4 - 2 = 2.

  4. 4

    Avoid this incomplete answer

    A student may write only x = 5 for the intersection, but a point on a plane needs both x and y coordinates.

Definition

The intersection of two linear graphs is the point where the two lines meet. Algebraically, it represents the common solution of the two linear equations.

Example

The lines x + y = 6 and x - y = 2 intersect at (4, 2) because 4 + 2 = 6 and 4 - 2 = 2.

Rule to remember

Common solution rule: if (x, y) lies on both lines, it satisfies both linear equations.

Memory hook

Where lines meet, equations agree.

Examples and method

Worked example

Solve x + y = 9 and x - y = 1. Adding gives 2x = 10, so x = 5. Then y = 4. Graphically, the two lines intersect at (5, 4).

Method to apply

Draw or solve both equations, find the common ordered pair, check it in both equations, and state it as the intersection point.

Diagram support

Draw both straight lines on the same coordinate plane and mark the crossing point with its ordered-pair coordinates.

How CBSE asks it

Students may be asked to find the point of intersection algebraically, identify it from a graph, or explain its meaning as a common solution.

Avoid common mistakes

Common confusion

Students sometimes read the crossing point from the graph but forget to write it as an ordered pair.

Common wrong answer

A student may write only x = 5 for the intersection, but a point on a plane needs both x and y coordinates.

Exam tip

Always verify the intersection point in both equations, especially if the graph scale is small.

Quick check

What does the intersection point of x + y = 8 and x - y = 2 represent?

It represents the ordered pair that satisfies both equations at the same time; solving gives (5, 3).

Study the intersection of two linear polynomials diagram carefully

Use the labelled diagram to keep intersection of two linear polynomials clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of intersection of two linear polynomials in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on intersection of two linear polynomials.

Revision cue

Revise intersection of two linear polynomials through the labels before writing the answer.

Answer writing and exam use

1-mark answer

The intersection of two linear graphs is the point where the two lines meet. Algebraically, it represents the common solution of the two linear equations.

2-mark answer

The intersection of two linear graphs is the point where the two lines meet. Algebraically, it represents the common solution of the two linear equations. Common solution rule: if (x, y) lies on both lines, it satisfies both linear equations. The lines x + y = 6 and x - y = 2 intersect at (4, 2) because 4 + 2 = 6 and 4 - 2 = 2.

3-mark answer

When two straight lines meet at one point, the coordinates of that point satisfy both equations. Solving the pair of equations gives the same ordered pair as the intersection point on the graph. This connects algebraic solving with graphical meaning. Common solution rule: if (x, y) lies on both lines, it satisfies both linear equations. Solve x + y = 9 and x - y = 1. Adding gives 2x = 10, so x = 5. Then y = 4. Graphically, the two lines intersect at (5, 4). Students may be asked to find the point of intersection algebraically, identify it from a graph, or explain its meaning as a common solution. A student may write only x = 5 for the intersection, but a point on a plane needs both x and y coordinates.
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