5.1 Variables and Algebraic Expressions
1. Concept of Variables
A variable is a quantity whose value is unknown or can change depending on the context.
- Fixed Value: A variable has a fixed value if the quantity it represents does not change in a given situation.
- Example: The distance between Kuala Lumpur and Penang ($d$).
- Varied Value: A variable has a varied value if the quantity it represents changes over time or across different scenarios.
- Example: The daily temperature of a city ($t$).
2. Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and mathematical operational symbols ($+$, $-$, $\times$, $\div$).
- Example: $3x + 5y - 7$
- Evaluating Expressions: Substitute given values for the variables into the expression and compute using standard order of operations.
- Example: If $x = 2$ and $y = 3$, then $3(2) + 5(3) - 7 = 6 + 15 - 7 = 14$.
3. Terms and Coefficients
In an algebraic expression, terms are parts separated by addition or subtraction signs.
- In the expression $5x - 3y + 8$, the terms are $5x$, $-3y$, and $8$.
- Coefficient: The numerical factor multiplying the variable in a term.
- In the term $5x$, the coefficient of $x$ is $5$.
- In the term $-3y$, the coefficient of $y$ is $-3$.
- In the term $-xy$, the coefficient of $y$ is $-x$, and the coefficient of $x$ is $-y$.
- Like Terms vs. Unlike Terms:
- Like Terms: Terms containing the same variables raised to the same power (e.g., $3x$ and $-7x$, or $2ab$ and $5ba$).
- Unlike Terms: Terms with different variables or different powers (e.g., $3x$ and $3y$, or $4x$ and $4x^2$).
5.2 Algebraic Expressions Involving Basic Arithmetic Operations
1. Addition and Subtraction of Algebraic Expressions
Only like terms can be added or subtracted by operating on their numerical coefficients.
- Example 1: $5x + 3x = (5 + 3)x = 8x$
- Example 2: $(3x + 4y) - (x - 2y) = 3x + 4y - x + 2y = (3x - x) + (4y + 2y) = 2x + 6y$
2. Multiplication and Division of Algebraic Expressions
Unlike terms can be multiplied and divided. Variables are multiplied or divided using index rules.
- Multiplication: Multiply numerical coefficients together, and combine repeated variable factors using powers.
- Example: $3a \times 4b = (3 \times 4) \times a \times b = 12ab$
- Example: $2x \times 5x^2 = 10x^3$
- Division: Divide numerical coefficients and cancel out common variable factors in numerator and denominator.
- Example: $12a^2b \div 3ab = \frac{12 \times a \times a \times b}{3 \times a \times b} = 4a$