Algebraic Expressions — The Building Blocks of Algebra
"Algebraic expressions use LETTERS to represent NUMBERS. This lets us write GENERAL rules that work for ANY value."
1. What Are Algebraic Expressions?
An algebraic expression is a combination of CONSTANTS and VARIABLES connected by arithmetic operations (+, −, ×, ÷).
Examples: 3x + 5, 2a − 7b + 4, x² + 2xy + y², 4p/q + 3.
Constants and Variables
| Type | Description | Examples |
|---|---|---|
| Constant | Fixed value — does NOT change | 5, −3, ½, π |
| Variable | Can take DIFFERENT values | x, y, a, b, p, q |
'Constants are the FIXED numbers. Variables are the LETTERS that can change. An algebraic expression is like a recipe — the variables are the ingredients that can vary.'
2. Terms, Factors, and Coefficients
Terms
Term: A part of an expression SEPARATED by + or − signs.
| Expression | Terms |
|---|---|
| 3x + 5y − 7 | Three terms: 3x, 5y, −7 |
| 4ab − 3c + 2d + 9 | Four terms: 4ab, −3c, 2d, 9 |
| 7p²q | One term (monomial) |
Factors and Coefficients
In the term −5xy, the factors are −5, x, and y. The numerical coefficient (or just coefficient) is the NUMERICAL factor: −5.
| Term | Numerical Coefficient | Variable Factors |
|---|---|---|
| 7x | 7 | x |
| −3ab | −3 | a, b |
| x²y | 1 | x², y |
| −p | −1 | p |
'When no number is written, the coefficient is 1 (for positive) or −1 (for negative). x means 1x. −y means −1y.'
3. Types of Algebraic Expressions
| Type | Number of Terms | Examples |
|---|---|---|
| Monomial | ONE | 5x, −7ab, 3p²q, 9 |
| Binomial | TWO | 2x + 3, a² − b², 4p + 7q |
| Trinomial | THREE | x + y + z, 2a² − 3ab + b² |
| Polynomial | MANY (any number) | Any of the above |
Note: 'Every monomial, binomial, and trinomial is also a POLYNOMIAL. Polynomial means "many terms" but is used as a general term.'
4. Like and Unlike Terms
Like terms: Terms with the SAME variable factors raised to the SAME powers. Unlike terms: Terms with DIFFERENT variable factors.
| Term 1 | Term 2 | Like? | Reason |
|---|---|---|---|
| 3x | 5x | YES | Same variable x |
| 4ab | −2ab | YES | Same variables a, b |
| 2x² | 5x | NO | Powers differ (x² vs x) |
| 7xy | 3yx | YES | xy = yx (commutative) |
| 5a²b | −2ab² | NO | Powers differ (a²b vs ab²) |
'Only LIKE terms can be added or subtracted. Unlike terms are left as they are.'
5. Addition and Subtraction of Algebraic Expressions
Adding/Subtracting Like Terms
'Add or subtract the COEFFICIENTS. Keep the variable part the SAME.'
Examples:
- 3x + 5x = (3 + 5)x = 8x
- 7ab − 3ab = 4ab
- −2p²q + 5p²q = 3p²q
Adding Expressions — Horizontal Method
(3x + 5y − 2) + (4x − 3y + 7) = 3x + 5y − 2 + 4x − 3y + 7 = (3x + 4x) + (5y − 3y) + (−2 + 7) = 7x + 2y + 5
Adding Expressions — Column Method
'Arrange like terms in VERTICAL columns, then add.'
3x + 5y − 2
+ 4x − 3y + 7
————————————
7x + 2y + 5
Subtracting Expressions
'Subtraction means adding the NEGATIVE. Change the sign of EVERY term in the second expression, then add.'
Example: Subtract (3x + 2y − 5) from (7x − 3y + 4). = (7x − 3y + 4) − (3x + 2y − 5) = 7x − 3y + 4 − 3x − 2y + 5 = 4x − 5y + 9
Column Method for Subtraction
7x − 3y + 4
− 3x + 2y − 5
————————————
4x − 5y + 9
6. Finding the Value of an Expression
Steps: 1. Substitute the given value of the variable. 2. Simplify using BODMAS/order of operations.
Example 1: Find the value of 3x + 5 when x = 4. 3(4) + 5 = 12 + 5 = 17.
Example 2: Find the value of 2a² − 3ab + b² when a = 5 and b = 3. 2(5)² − 3(5)(3) + (3)² = 2(25) − 45 + 9 = 50 − 45 + 9 = 14.
Example 3: 'The perimeter of a rectangle is 2(l + b). Find the perimeter when l = 12 and b = 8.' P = 2(12 + 8) = 2(20) = 40 units.
Evaluating Expressions with Negative Values
Example: Find 2x² − 5x + 3 when x = −2. 2(−2)² − 5(−2) + 3 = 2(4) + 10 + 3 = 8 + 10 + 3 = 21.
7. Using Algebraic Expressions in Formulas
'Many formulas from Class 7 are algebraic expressions:'
| Formula | As Algebraic Expression |
|---|---|
| Area of rectangle = l × b | A = lb |
| Perimeter of square = 4 × side | P = 4s |
| Simple Interest = (P × R × T)/100 | SI = PRT/100 |
| Speed = Distance/Time | s = d/t |
8. Common Mistakes and Fixes
| Mistake | Why It Is Wrong | Correct Approach |
|---|---|---|
| 3x + 5y = 8xy | Cannot add UNLIKE terms | 3x + 5y stays as is |
| 2x + 3x² = 5x³ | Different powers means unlike | 2x + 3x² stays as is |
| Subtracting: (5x+3) − (2x+1) = 5x+3−2x+1 | Forgot to change ALL signs | 5x+3−2x−1 = 3x+2 |
| 3x means 3 + x | 3x means 3 × x | 3x = 3 × x, not 3 + x |
| Coefficient of x is 0 | If written as just x, coefficient = 1 | x = 1x, −x = −1x |
9. AP SSC Exam Focus
| Topic | Marks | Question Type |
|---|---|---|
| Identifying terms, factors, coefficients | 2-3 | Define and identify |
| Like and unlike terms | 2-3 | Group and classify |
| Addition and subtraction | 3-4 | Simplify expressions |
| Finding value of expression | 3-4 | Substitution problems |
Quick Self-Test
Q1. Identify the terms in: 4x² − 3xy + 7y² − 5. A1. Terms: 4x², −3xy, 7y², −5.
Q2. Are 3x²y and −5xy² like terms? A2. No. 3x²y has x²y while −5xy² has xy². Powers differ.
Q3. Simplify: (7a + 3b − 5) + (2a − 8b + 9). A3. 7a + 3b − 5 + 2a − 8b + 9 = 9a − 5b + 4.
Q4. Subtract (5p − 3q + 2) from (8p + q − 7). A4. (8p + q − 7) − (5p − 3q + 2) = 8p + q − 7 − 5p + 3q − 2 = 3p + 4q − 9.
Q5. Find the value of 2x² − 3x + 1 when x = 5. A5. 2(25) − 3(5) + 1 = 50 − 15 + 1 = 36.
Q6. Find the value of a² + b² − 2ab when a = 7, b = 3. A6. 49 + 9 − 2(21) = 49 + 9 − 42 = 16.
Q7. What is the numerical coefficient of −7x²yz? A7. −7.
