Algebra
Algebraic Expressions, Identities and Linear Equations
Overview
Algebra forms the backbone of secondary mathematics and appears consistently in Assam TET Paper II. This topic tests your ability to manipulate symbols, simplify expressions, apply standard identities quickly, and solve equations—skills essential for any mathematics teacher working with classes VI to VIII.
For the exam, expect questions that combine multiple concepts: simplifying an expression using an identity, then solving for a variable. Mastery here also supports success in later topics like quadratic equations and geometry (where algebraic methods prove lengths or areas). Students who internalize the identities and practice systematic equation-solving will find this section high-scoring.
The key is pattern recognition. Once you see which identity applies or how to isolate the variable, the arithmetic becomes straightforward. Focus on speed and accuracy—errors in sign handling or transposition are the most common traps.
Key Concepts
- Algebraic Expression: A combination of constants, variables and operations (e.g., 3x² + 2xy − 5). No equals sign—that makes it an equation.
- Terms, Coefficients and Degree: In 4x³y, the coefficient is 4, variables are x and y, and the degree is 3 + 1 = 4. Degree of a polynomial is the highest sum of exponents in any term.
- Like and Unlike Terms: Like terms share identical variable parts (3xy and −7xy). Only like terms can be added or subtracted directly.
- Polynomial Classification: Monomial (1 term), binomial (2 terms), trinomial (3 terms). Degree-based: linear (degree 1), quadratic (degree 2), cubic (degree 3).
- Algebraic Identity: An equation true for all values of variables. Identities enable quick expansion and factorization without long multiplication.
- Linear Equation in One Variable: Form ax + b = 0. Solution involves isolating x through inverse operations while maintaining equality.
- Linear Equation in Two Variables: Form ax + by + c = 0. Infinite solutions lying on a straight line; unique solution found when two such equations intersect.
Formulas / Key Facts
Standard Algebraic Identities
| Identity | Expanded Form |
|---|---|
| (a + b)² | a² + 2ab + b² |
| (a − b)² | a² − 2ab + b² |
| (a + b)(a − b) | a² − b² |
| (a + b)³ | a³ + 3a²b + 3ab² + b³ = a³ + b³ + 3ab(a + b) |
| (a − b)³ | a³ − 3a²b + 3ab² − b³ = a³ − b³ − 3ab(a − b) |
| a³ + b³ | (a + b)(a² − ab + b²) |
| a³ − b³ | (a − b)(a² + ab + b²) |
| (a + b + c)² | a² + b² + c² + 2ab + 2bc + 2ca |
Linear Equation Facts
- Transposition rule: Moving a term across the equals sign reverses its sign.
- Cross-multiplication: For a/b = c/d, we get ad = bc.
- Solution of ax + b = 0: x = −b/a (provided a ≠ 0).
- Simultaneous equations (two variables): Use substitution or elimination to find unique (x, y).
Worked Examples
Example 1: Simplify using identity
Problem: Evaluate 103² without direct multiplication.
Solution: Write 103 as (100 + 3). Apply (a + b)² = a² + 2ab + b². = 100² + 2 × 100 × 3 + 3² = 10000 + 600 + 9 = 10609
Example 2: Factorize using identity
Problem: Factorize 4x² − 9y².
Solution: Recognize as difference of squares: a² − b² = (a + b)(a − b). Here a² = 4x² → a = 2x; b² = 9y² → b = 3y. = (2x + 3y)(2x − 3y)
Example 3: Solve linear equation
Problem: Solve 5(x − 2) + 3 = 2(x + 4).
Solution: Step 1: Expand both sides. 5x − 10 + 3 = 2x + 8 5x − 7 = 2x + 8
Step 2: Bring variable terms to one side, constants to other. 5x − 2x = 8 + 7 3x = 15
Step 3: Divide. x = 15/3 = 5
Verification: LHS = 5(5 − 2) + 3 = 15 + 3 = 18; RHS = 2(5 + 4) = 18 ✓
Example 4: Simultaneous equations (elimination)
Problem: Solve 2x + 3y = 12 and 4x − y = 5.
Solution: Multiply second equation by 3: 12x − 3y = 15. Add to first equation: 2x + 3y + 12x − 3y = 12 + 15 14x = 27 x = 27/14
Substitute in 4x − y = 5: 4(27/14) − y = 5 108/14 − y = 5 y = 54/7 − 5 = 54/7 − 35/7 = 19/7
Answer: x = 27/14, y = 19/7
Common Mistakes
| Wrong Thinking | Correct Fix |
|---|---|
| Squaring a binomial incorrectly: (a + b)² = a² + b² | Remember the middle term: (a + b)² = a² + 2ab + b². Never skip 2ab. |
| Sign errors during transposition: Moving +5 to other side as +5 | When a term crosses the equals sign, its sign flips: +5 becomes −5. |
| Confusing identity direction: Trying to use a² − b² when expression is a² + b² | a² + b² has no simple factorization over real numbers. Recognize when identity applies and when it does not. |
| Dividing only part of the equation: Dividing just one side by a number | Any operation must be applied to the entire LHS and entire RHS to maintain equality. |
| Dropping terms when expanding (a + b + c)²: Forgetting 2bc or 2ca | Use the formula systematically: three squares plus three double-products. |
Quick Reference
- (a + b)² = a² + 2ab + b² — always include the middle term.
- a² − b² = (a + b)(a − b) — fastest factorization tool.
- Degree of polynomial = highest sum of exponents in any term.
- Transpose with sign change — moving term across "=" flips its sign.
- Verify solutions — substitute back to catch arithmetic slips.
- Elimination/Substitution — two equations, two unknowns → unique solution.