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Pair of linear equations in two variables

A pair of linear equations in two variables is two first-degree equations written together, such as ax + by + c = 0 and px + qy + r = 0, with the same two unknowns.

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Student-friendly explanation

This chapter studies two equations together because many real problems need two conditions to be satisfied at the same time. The solution is the ordered pair of values of x and y that makes both equations true. In exams, students must learn to identify the type of pair and choose a suitable method quickly.

How to write this in exams

  1. 1

    Start with the exact idea

    A pair of linear equations in two variables is two first-degree equations written together, such as ax + by + c = 0 and px + qy + r = 0, with the same two unknowns.

  2. 2

    Then show how to use it

    1. Write both equations in standard form if needed. 2. Decide whether substitution, elimination, or graphing is suitable. 3. Solve for one variable. 4. Substitute back to get the other variable. 5. Verify the answer in both equations.

  3. 3

    Add one concrete example

    For example, 2x + y = 7 and x - y = 1 form a pair of linear equations in two variables. Their common solution is the pair that satisfies both equations.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to find only one variable and stop there. That is incomplete because a pair solution needs both x and y.

Definition

A pair of linear equations in two variables is two first-degree equations written together, such as ax + by + c = 0 and px + qy + r = 0, with the same two unknowns.

Example

For example, 2x + y = 7 and x - y = 1 form a pair of linear equations in two variables. Their common solution is the pair that satisfies both equations.

Rule to remember

A linear equation in two variables is usually written as ax + by + c = 0, where a and b are not both zero.

Memory hook

Think of the two equations as two rules and the solution as the one pair that obeys both rules together.

Examples and method

Worked example

For 2x + y = 7 and x - y = 1, add the equations after rewriting if needed: 2x + y = 7 and x - y = 1. Adding gives 3x = 8, so x = 8/3. Substituting in x - y = 1 gives y = 5/3.

Method to apply

1. Write both equations in standard form if needed. 2. Decide whether substitution, elimination, or graphing is suitable. 3. Solve for one variable. 4. Substitute back to get the other variable. 5. Verify the answer in both equations.

Diagram support

A graph of two linear equations shows two straight lines on the same coordinate plane, and their meeting point or overlap tells the solution type.

How CBSE asks it

The exam may ask you to classify a pair, solve it by a method, or identify the ordered pair that satisfies both equations.

Avoid common mistakes

Common confusion

Many students treat the two equations separately and write two different answers. That is wrong because the correct answer must satisfy both equations together.

Common wrong answer

A common wrong answer is to find only one variable and stop there. That is incomplete because a pair solution needs both x and y.

Exam tip

Always write both equations clearly first, then decide whether graphing, substitution, or elimination will be fastest for the given question.

Quick check

If two linear equations are given together, what must the final solution satisfy?

The final solution must satisfy both linear equations at the same time. In other words, the same values of x and y should make each equation true together.

Study the pair of linear equations in two variables diagram carefully

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

What this diagram makes clear

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

Where this helps in exams

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

Revision cue

Revise pair of linear equations in two variables through the labels before writing the answer.

Answer writing and exam use

1-mark answer

A pair of linear equations in two variables is two first-degree equations written together, such as ax + by + c = 0 and px + qy + r = 0, with the same two unknowns.

2-mark answer

A pair of linear equations in two variables is two first-degree equations written together, such as ax + by + c = 0 and px + qy + r = 0, with the same two unknowns. A linear equation in two variables is usually written as ax + by + c = 0, where a and b are not both zero. For example, 2x + y = 7 and x - y = 1 form a pair of linear equations in two variables. Their common solution is the pair that satisfies both equations.

3-mark answer

This chapter studies two equations together because many real problems need two conditions to be satisfied at the same time. The solution is the ordered pair of values of x and y that makes both equations true. In exams, students must learn to identify the type of pair and choose a suitable method quickly. A linear equation in two variables is usually written as ax + by + c = 0, where a and b are not both zero. For 2x + y = 7 and x - y = 1, add the equations after rewriting if needed: 2x + y = 7 and x - y = 1. Adding gives 3x = 8, so x = 8/3. Substituting in x - y = 1 gives y = 5/3. The exam may ask you to classify a pair, solve it by a method, or identify the ordered pair that satisfies both equations. A common wrong answer is to find only one variable and stop there. That is incomplete because a pair solution needs both x and y.
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