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Solving Linear Polynomials

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Student-friendly explanation

How to write this in exams

1
Start with the exact idea
Solving linear polynomials means finding the value or values of variables that make the given linear equation true. At this level, substitution, simple elimination, and verification are useful methods.
2
Then show how to use it
Align equations, decide whether addition or subtraction cancels a variable, solve for one variable, substitute back to find the other, and verify both original equations.
3
Add one concrete example
To solve x + y = 9 and x - y = 1, add the equations to get 2x = 10, so x = 5. Then 5 + y = 9 gives y = 4.
4
Avoid this incomplete answer
For x + y = 7 and x - y = 3, a wrong answer is y = 5 if the student adds y and -y as 2y instead of 0.

Definition

Example

To solve x + y = 9 and x - y = 1, add the equations to get 2x = 10, so x = 5. Then 5 + y = 9 gives y = 4.

Rule to remember

Elimination idea: add or subtract equations so that one variable cancels. Substitution idea: express one variable in terms of the other and replace it in the second equation.

Memory hook

Elimination means one variable should disappear from the working step, not from the problem meaning.

Examples and method

Worked example

Solve 2x + y = 11 and x + y = 7. Subtract the second equation from the first: x = 4. Put x = 4 in x + y = 7, so y = 3. Solution is (4, 3).

Method to apply

Align equations, decide whether addition or subtraction cancels a variable, solve for one variable, substitute back to find the other, and verify both original equations.

Diagram support

A simple flow chart can show: choose method, remove one variable, find first value, find second value, verify.

How CBSE asks it

Students may be asked to solve a pair, verify a claimed solution, or identify an error in a student's elimination step.

Avoid common mistakes

Common confusion

Students often eliminate the wrong terms or forget to change signs when subtracting equations.

Common wrong answer

For x + y = 7 and x - y = 3, a wrong answer is y = 5 if the student adds y and -y as 2y instead of 0.

Exam tip

After solving, substitute the values in the original equation or equations, not only in your last working line.

Quick check

Solve x + y = 7 and x - y = 3 using elimination.

Adding the equations gives 2x = 10, so x = 5. Substituting in x + y = 7 gives y = 2, so the solution is (5, 2).

Answer writing and exam use

1-mark answer

2-mark answer

Solving linear polynomials means finding the value or values of variables that make the given linear equation true. At this level, substitution, simple elimination, and verification are useful methods. Elimination idea: add or subtract equations so that one variable cancels. Substitution idea: express one variable in terms of the other and replace it in the second equation. To solve x + y = 9 and x - y = 1, add the equations to get 2x = 10, so x = 5. Then 5 + y = 9 gives y = 4.

3-mark answer

For one variable, solving usually means isolating the variable. For two variables, one equation can have many solutions, while two linear equations may be solved together by substitution or elimination. Verification is important because it catches sign and arithmetic mistakes. Elimination idea: add or subtract equations so that one variable cancels. Substitution idea: express one variable in terms of the other and replace it in the second equation. Solve 2x + y = 11 and x + y = 7. Subtract the second equation from the first: x = 4. Put x = 4 in x + y = 7, so y = 3. Solution is (4, 3). Students may be asked to solve a pair, verify a claimed solution, or identify an error in a student's elimination step. For x + y = 7 and x - y = 3, a wrong answer is y = 5 if the student adds y and -y as 2y instead of 0.

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