Linear equations, expressions, identities and factorisation
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Overview
Algebra forms the bridge between arithmetic and higher mathematics, introducing students to the power of symbols and generalisation. For PSTET Paper II, algebra questions test both your conceptual understanding and your ability to teach these abstract ideas to Classes VI-VIII students. Expect questions on simplifying expressions, solving linear equations, applying standard identities, and factorising polynomials.
This topic carries significant weight because it underpins problem-solving across mathematics. Examiners often frame questions as word problems requiring equation formation, or ask you to identify errors in student work—testing pedagogical awareness alongside content mastery. A strong grip on identities and factorisation techniques is essential, as these appear repeatedly in both content and pedagogy sections.
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Key Concepts
**Algebraic Expression**: A combination of variables, constants, and operations (e.g., 3x + 5y − 7). Unlike equations, expressions have no equality sign.
**Terms, Coefficients, and Constants**: In 4x² − 3x + 2, there are three terms; 4 and −3 are coefficients; 2 is the constant term.
**Like and Unlike Terms**: Terms with identical variable parts (e.g., 5xy and −2xy) are like terms and can be combined; unlike terms cannot.
**Linear Equation in One Variable**: An equation of the form ax + b = 0 where a ≠ 0. The solution is x = −b/a.
**Linear Equation in Two Variables**: Form ax + by + c = 0. Represents a straight line; infinite solutions exist as ordered pairs (x, y).
**Algebraic Identity**: An equality true for all values of variables. Identities are used for expansion and factorisation—not to be confused with equations that hold only for specific values.
**Factorisation**: Expressing a polynomial as a product of its factors. Reverse of expansion.
**Polynomial Degree**: The highest power of the variable. Degree 1 = linear, Degree 2 = quadratic.
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Formulas / Key Facts
### Standard Algebraic Identities (Class VIII)
| Identity | Expanded Form | |----------|---------------| | (a + b)² | a² + 2ab + b² | | (a − b)² | a² − 2ab + b² | | (a + b)(a − b) | a² − b² | | (x + a)(x + b) | x² + (a + b)x + ab | | (a + b + c)² | a² + b² + c² + 2ab + 2bc + 2ca |
### Solving Linear Equations
**Transposition Rule**: When a term moves across the equals sign, its sign changes.
**Cross-multiplication** (for equations with fractions): If a/b = c/d, then ad = bc.
**Solution**: Using (a + b)² = a² + 2ab + b² where a = 2x and b = 3y
= (2x)² + 2(2x)(3y) + (3y)² = 4x² + 12xy + 9y²
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### Example 3: Factorisation by Splitting Middle Term
**Problem**: Factorise x² − 7x + 12
**Solution**: Step 1: Find two numbers whose product = 12 and sum = −7 Numbers: −3 and −4 (since −3 × −4 = 12 and −3 + −4 = −7)
Step 2: Split middle term x² − 3x − 4x + 12
Step 3: Group and factor x(x − 3) − 4(x − 3) = (x − 4)(x − 3)
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Common Mistakes
| Wrong Thinking | Correct Fix | |----------------|-------------| | **Confusing expressions with equations**: Students try to "solve" 3x + 5 by setting it equal to zero. | Clarify that expressions are simplified, not solved. Only equations (with = sign) are solved. | | **Sign errors during transposition**: Moving +5 to the other side and keeping it +5. | Emphasise: crossing the equals sign always flips the sign. Use physical balance analogy. | | **Misapplying identities**: Writing (a + b)² = a² + b², forgetting the middle term. | Drill the complete identity. Have students verify with numbers: (2 + 3)² = 25 ≠ 4 + 9. | | **Incorrect grouping in factorisation**: Grouping terms that don't share common factors. | Teach students to check that each group yields the same bracket factor before proceeding. | | **Mixing coefficient and constant**: Treating the coefficient 3 in 3x as a separate term. | Reinforce that 3x is one term; the coefficient 3 is attached to variable x. |
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Quick Reference
**Expression** = no equals sign; **Equation** = has equals sign with solution(s).
**(a + b)² = a² + 2ab + b²** — never forget the middle term 2ab.
**a² − b² = (a + b)(a − b)** — the difference-of-squares identity, very common in factorisation.
**Transposition flips signs**: +5 becomes −5 when moved across =.
**Splitting middle term**: Find two numbers with required product and sum to factorise quadratics.
For **linear equation in one variable**: isolate variable → one unique solution.
👥 Study this together
Invite your prep group — read the same notes, then discuss doubts in this topic's shared room.