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Linear Polynomial in Two Variables

A linear polynomial or linear equation in two variables has the form ax + by + c = 0, where x and y are variables, a, b, and c are constants, and a and b are not both zero.

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Student-friendly explanation

A solution of a linear equation in two variables is an ordered pair (x, y) that makes the equation true. Unlike a one-variable linear equation, a two-variable linear equation usually has many solutions, and these solutions form a straight line on a graph.

How to write this in exams

  1. 1

    Start with the exact idea

    A linear polynomial or linear equation in two variables has the form ax + by + c = 0, where x and y are variables, a, b, and c are constants, and a and b are not both zero.

  2. 2

    Then show how to use it

    Take the ordered pair, substitute the first value for x and the second for y, simplify both sides, and conclude whether the equation is satisfied.

  3. 3

    Add one concrete example

    For x + y = 6, the ordered pairs (1, 5), (2, 4), and (6, 0) are solutions because each pair has x + y equal to 6.

  4. 4

    Avoid this incomplete answer

    For (2, 5), students may substitute x = 5 and y = 2 by reversing the order, leading to a wrong conclusion.

Definition

A linear polynomial or linear equation in two variables has the form ax + by + c = 0, where x and y are variables, a, b, and c are constants, and a and b are not both zero.

Example

For x + y = 6, the ordered pairs (1, 5), (2, 4), and (6, 0) are solutions because each pair has x + y equal to 6.

Rule to remember

Standard form: ax + by + c = 0. A solution is an ordered pair (x, y) satisfying the equation.

Memory hook

Ordered pair means order matters: x first, y second.

Examples and method

Worked example

Check whether (4, -1) satisfies x - 2y = 6. Substitute x = 4 and y = -1: 4 - 2(-1) = 4 + 2 = 6. So it is a solution.

Method to apply

Take the ordered pair, substitute the first value for x and the second for y, simplify both sides, and conclude whether the equation is satisfied.

Diagram support

A two-column table of x and y values can be used before plotting points on the coordinate plane.

How CBSE asks it

Students may be asked to verify an ordered pair, complete a value table, or generate two solutions for an equation.

Avoid common mistakes

Common confusion

Students sometimes write only one number as the answer. In two variables, the answer must be an ordered pair like (2, 4).

Common wrong answer

For (2, 5), students may substitute x = 5 and y = 2 by reversing the order, leading to a wrong conclusion.

Exam tip

Substitute x and y in the correct order. The first coordinate is x and the second coordinate is y.

Quick check

Is (3, 2) a solution of 2x + y = 8?

Yes, (3, 2) is a solution because 2(3) + 2 = 6 + 2 = 8, which satisfies the equation.

Study the linear polynomial in two variables diagram carefully

Use the labelled diagram to keep linear polynomial in two variables clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of linear polynomial in two variables in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on linear polynomial in two variables.

Revision cue

Revise linear polynomial in two variables through the labels before writing the answer.

Answer writing and exam use

1-mark answer

A linear polynomial or linear equation in two variables has the form ax + by + c = 0, where x and y are variables, a, b, and c are constants, and a and b are not both zero.

2-mark answer

A linear polynomial or linear equation in two variables has the form ax + by + c = 0, where x and y are variables, a, b, and c are constants, and a and b are not both zero. Standard form: ax + by + c = 0. A solution is an ordered pair (x, y) satisfying the equation. For x + y = 6, the ordered pairs (1, 5), (2, 4), and (6, 0) are solutions because each pair has x + y equal to 6.

3-mark answer

A solution of a linear equation in two variables is an ordered pair (x, y) that makes the equation true. Unlike a one-variable linear equation, a two-variable linear equation usually has many solutions, and these solutions form a straight line on a graph. Standard form: ax + by + c = 0. A solution is an ordered pair (x, y) satisfying the equation. Check whether (4, -1) satisfies x - 2y = 6. Substitute x = 4 and y = -1: 4 - 2(-1) = 4 + 2 = 6. So it is a solution. Students may be asked to verify an ordered pair, complete a value table, or generate two solutions for an equation. For (2, 5), students may substitute x = 5 and y = 2 by reversing the order, leading to a wrong conclusion.
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