Algebraic Expressions
1. Variables and Constants
- A constant has a fixed value (e.g., 5, -3, 0.7).
- A variable is a symbol (usually a letter like x, y, z) that can take different values.
Example: In the expression 4x + 7, 4 and 7 are constants, and x is a variable.
2. Algebraic Expressions
An algebraic expression combines variables and constants using arithmetic operations.
Examples:
- 3x + 2
- 5y - 7
- 2a + 3b - 4
- x^2 + 2x + 1
Worked Example: Write an expression for '5 added to twice a number.'
Let the number be n. Twice the number = 2n. Five added = 2n + 5.
Common Mistake: Writing '3 more than x' as 3x. 'More than' means addition: x + 3, not multiplication.
3. Terms of an Expression
A term is a part of an expression separated by + or - signs.
In 3x^2 + 5x - 7, the terms are 3x^2, 5x, and -7.
Constant term: A term with no variable (like -7 in the above).
Coefficient: The numerical factor of a term.
In 3x^2, 3 is the coefficient of x^2.
In 5x, 5 is the coefficient of x.
Exam Focus (2 marks): 'In the expression 4x^2 - 7x + 9, identify the terms and their coefficients.'
Term 4x^2: coefficient 4. Term -7x: coefficient -7. Term 9: constant term.
4. Like and Unlike Terms
Like terms have the same variables raised to the same powers. Only the coefficients differ.
Unlike terms have different variables or the same variables with different powers.
| Like Terms | Unlike Terms |
|---|---|
| 3x, 5x, -2x | 3x, 3y (different variables) |
| 4a^2, -a^2, 7a^2 | 4a^2, 4a (different powers) |
| 2xy, -5xy, xy | 2xy, 2x (different variables) |
Worked Example: Identify like terms: 5xy, 3x, -2xy, 7x, 4y.
Like terms: 5xy and -2xy (both have xy). 3x and 7x (both have x). 4y has no like term.
5. Adding and Subtracting Like Terms
Only like terms can be added or subtracted. Add/subtract the coefficients, keep the variable part unchanged.
Worked Example: Simplify 3x + 5y - 2x + 3y.
3x - 2x = x. 5y + 3y = 8y. Answer = x + 8y.
Common Mistake: Trying to add unlike terms, e.g., writing 3x + 4y = 7xy. This is incorrect because the terms are unlike.
6. Multiplication of Algebraic Expressions
Numerical coefficients multiply. Variables multiply by adding exponents (for same variable).
Worked Example: 3x x 4y = (3 x 4) x (x x y) = 12xy.
Worked Example: 2a x 5a = (2 x 5) x (a x a) = 10a^2.
7. Finding the Value of an Expression
Substitute the given value for the variable and simplify.
Worked Example: Find the value of 3x^2 - 2x + 1 when x = 2.
3(2)^2 - 2(2) + 1 = 3(4) - 4 + 1 = 12 - 4 + 1 = 9.
Exam Focus (3 marks): 'Evaluate 2a + 3b when a = 4 and b = -2.'
2(4) + 3(-2) = 8 - 6 = 2.
8. Simple Equations by Trial
Find the value of the variable that makes the equation true.
Worked Example: Solve x + 5 = 12 by trial.
Try x = 5: 5 + 5 = 10 (not 12).
Try x = 7: 7 + 5 = 12 (correct!).
Solution: x = 7.
9. Comparison Table: Expression vs Equation
| Aspect | Expression | Equation |
|---|---|---|
| Contains | Variables, constants, operations | Variables, constants, operations, = |
| Example | 3x + 5 | 3x + 5 = 14 |
| Value | Depends on variable | True for specific value(s) |
| Solve | Not applicable | Find the variable value |
10. Self-Test
- Write an expression: '7 subtracted from thrice a number.'
- Identify like terms: 4ab, 3a, -2ba, 7b, ab.
- Simplify: 6p - 2q + 3p + 5q.
- Multiply: 4x x 7y.
- Find the value: 3m - 2n + 5, when m = 4 and n = -1.
- Solve by trial: 2y - 3 = 9.
- In 5x^2 - 3x + 8, what is: (a) the coefficient of x^2? (b) the constant term?
11. Answers to Self-Test
- Let the number be n. Thrice = 3n. Seven subtracted = 3n - 7.
- 4ab, -2ba (= -2ab), and ab are like terms. 3a and 7b have no like terms.
- 6p + 3p = 9p. -2q + 5q = 3q. Answer = 9p + 3q.
- (4 x 7) x (x x y) = 28xy.
- 3(4) - 2(-1) + 5 = 12 + 2 + 5 = 19.
- Try y = 5: 2(5) - 3 = 7 (not 9). Try y = 6: 2(6) - 3 = 9. y = 6.
- (a) 5 (b) 8.
