Algebraic Expressions, Identities and Linear Equations
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Overview
Algebra forms the backbone of upper-primary mathematics and carries significant weight in WB TET Paper II. This topic bridges arithmetic and higher mathematics by introducing symbolic representation of numbers and relationships. Students transitioning from Classes 6–8 must master how to form, simplify and manipulate algebraic expressions before they can solve equations confidently.
For the WB TET exam, expect questions on identifying terms, coefficients and like terms; applying standard algebraic identities for quick computation; and solving linear equations in one and two variables. Pedagogy questions often ask how to introduce abstract algebraic concepts to young learners using concrete examples. A firm grasp here also supports later topics in geometry (coordinate geometry) and mensuration (formula manipulation).
Candidates should focus on the four standard identities, the distinction between expressions and equations, and systematic methods for solving linear equations—these appear most frequently.
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Key Concepts
**Algebraic Expression**: A combination of constants, variables and operations (no equality sign). Example: 3x + 5y − 7.
**Terms, Coefficients and Constants**: In 4x² − 3x + 2, the terms are 4x², −3x and 2; the coefficient of x² is 4; the constant term is 2.
**Like and Unlike Terms**: Like terms have identical variable parts (3xy and −5xy); unlike terms differ in variables or powers (2x and 2x²).
**Polynomial**: An expression with non-negative integer exponents. Classified by degree—linear (degree 1), quadratic (degree 2), cubic (degree 3).
**Identity vs Equation**: An identity holds true for all values of variables; an equation is true only for specific values (the solutions).
**Linear Equation in One Variable**: Form ax + b = 0 (a ≠ 0). Exactly one solution: x = −b/a.
**Linear Equation in Two Variables**: Form ax + by + c = 0. Infinite solutions forming a straight line; a unique solution exists when paired with another such equation.
**Solving Simultaneous Equations**: Use substitution or elimination to find the unique (x, y) satisfying both equations.
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Formulas / Key Facts
### Standard Algebraic Identities
1. **(a + b)² = a² + 2ab + b²** — Square of a sum. 2. **(a − b)² = a² − 2ab + b²** — Square of a difference. 3. **(a + b)(a − b) = a² − b²** — Difference of squares. 4. **(x + a)(x + b) = x² + (a + b)x + ab** — Product of two binomials with common variable.
Substitute x in (ii): 4(27/14) − y = 5 108/14 − y = 5 54/7 − y = 5 y = 54/7 − 35/7 = 19/7
**Answer: x = 27/14, y = 19/7**
(In exams, values are usually integers; the method remains the same.)
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Common Mistakes
1. **Sign errors during expansion** *Wrong*: (a − b)² written as a² − b². *Fix*: Always remember the middle term: a² **− 2ab** + b².
2. **Confusing expression with equation** *Wrong*: Trying to "solve" 3x + 5 when there is no equality. *Fix*: Expressions are simplified, not solved. Equations (with '=') are solved.
3. **Ignoring the coefficient when transposing** *Wrong*: From 3x = 12, writing x = 12. *Fix*: Divide both sides by the coefficient—x = 12/3 = 4.
4. **Adding unlike terms** *Wrong*: 2x + 3x² = 5x³. *Fix*: Only like terms (same variable and power) combine. 2x + 3x² remains unchanged.
5. **Dropping a variable in simultaneous equations** *Wrong*: Solving one equation and assuming the other is satisfied. *Fix*: Always substitute back to verify both equations hold.
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Quick Reference
**Identity (a + b)² = a² + 2ab + b²** — memorise middle term sign.
**Difference of squares: a² − b² = (a + b)(a − b)** — useful for mental calculation.
**Linear equation one variable**: isolate variable → transpose → divide by coefficient.
**Simultaneous equations**: elimination (equalise coefficients) or substitution (express one variable).
**Degree of polynomial** = highest power of the variable.
**Coefficient** = numerical factor attached to a variable term.
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If 3x + 7 = 22, then what is the value of x?
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