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Algebraic solution by elimination

The elimination method solves a pair by making one variable cancel when the equations are added or subtracted.

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Student-friendly explanation

This method is useful when the coefficients of one variable are already equal or can be made equal by multiplying one or both equations. After elimination, only one variable remains, and we solve it first. Then we substitute back to get the second variable.

How to write this in exams

  1. 1

    Start with the exact idea

    The elimination method solves a pair by making one variable cancel when the equations are added or subtracted.

  2. 2

    Then show how to use it

    1. Look for a variable with equal or opposite coefficients. 2. If needed, multiply one or both equations to match coefficients. 3. Add or subtract to eliminate one variable. 4. Solve the remaining equation. 5. Substitute back to find the second variable.

  3. 3

    Add one concrete example

    For 2x + 3y = 13 and 2x - y = 5, subtracting the second from the first removes x immediately.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to multiply one equation correctly but then forget to multiply every term, including the constant term.

Definition

The elimination method solves a pair by making one variable cancel when the equations are added or subtracted.

Example

For 2x + 3y = 13 and 2x - y = 5, subtracting the second from the first removes x immediately.

Rule to remember

Make the coefficients of one variable equal if needed, then add or subtract the equations to eliminate that variable.

Memory hook

Match, cancel, solve, and return.

Examples and method

Worked example

For 3x + 2y = 16 and 3x - 4y = 4, subtract the second from the first: 6y = 12, so y = 2. Substitute into 3x + 2y = 16 to get 3x + 4 = 16, so x = 4.

Method to apply

1. Look for a variable with equal or opposite coefficients. 2. If needed, multiply one or both equations to match coefficients. 3. Add or subtract to eliminate one variable. 4. Solve the remaining equation. 5. Substitute back to find the second variable.

Diagram support

A small alignment table of coefficients can help decide whether addition or subtraction should be used.

How CBSE asks it

The exam may ask for the elimination method directly or for the first equation to be multiplied before elimination.

Avoid common mistakes

Common confusion

Students often add when subtraction is needed, or subtract when addition is needed, so the intended variable does not cancel.

Common wrong answer

A common wrong answer is to multiply one equation correctly but then forget to multiply every term, including the constant term.

Exam tip

Before operating on the equations, check which variable will cancel faster and choose the sign carefully.

Quick check

What is the main goal of elimination in a pair of linear equations?

The main goal is to cancel one variable so that only one variable remains. Then the simpler single-variable equation can be solved easily.

Answer writing and exam use

1-mark answer

The elimination method solves a pair by making one variable cancel when the equations are added or subtracted.

2-mark answer

The elimination method solves a pair by making one variable cancel when the equations are added or subtracted. Make the coefficients of one variable equal if needed, then add or subtract the equations to eliminate that variable. For 2x + 3y = 13 and 2x - y = 5, subtracting the second from the first removes x immediately.

3-mark answer

This method is useful when the coefficients of one variable are already equal or can be made equal by multiplying one or both equations. After elimination, only one variable remains, and we solve it first. Then we substitute back to get the second variable. Make the coefficients of one variable equal if needed, then add or subtract the equations to eliminate that variable. For 3x + 2y = 16 and 3x - 4y = 4, subtract the second from the first: 6y = 12, so y = 2. Substitute into 3x + 2y = 16 to get 3x + 4 = 16, so x = 4. The exam may ask for the elimination method directly or for the first equation to be multiplied before elimination. A common wrong answer is to multiply one equation correctly but then forget to multiply every term, including the constant term.
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