Making Sense of Algebra

1. Using Letters for Unknowns

In algebra, letters (variables) are used to represent numbers that are either unknown or can change.

Expression: A mathematical phrase without an equals sign (e.g., $3x + 5$).
Equation: A statement that two expressions are equal (e.g., $3x + 5 = 11$).
Formula: A rule showing the relationship between different variables (e.g., $A = lw$).

Substitution is the process of replacing variables with specific numerical values to evaluate an expression.

Example: If $x = 3$ and $y = -2$, evaluate $2x^2 - 3y$.
- $2(3)^2 - 3(-2)$
- $2(9) + 6 = 18 + 6 = 24$.
Key Rule: Always use brackets when substituting negative numbers to avoid sign errors.

Terms are “like” if they have the exact same variable part. Only like terms can be added or subtracted.

Example: $4a + 5b - 2a + 3b$
- Group like terms: $(4a - 2a) + (5b + 3b)$
- Simplify: $2a + 8b$.
Note: $x$ and $x^2$ are NOT like terms.

Multiply the term outside the bracket by every term inside the bracket.

Single Bracket: $a(b + c) = ab + ac$
Double Brackets (FOIL method): $(x + a)(x + b) = x^2 + bx + ax + ab$
- First, Outer, Inner, Last.

Factorisation is the inverse of expanding. It involves finding the Highest Common Factor (HCF) and placing it outside the bracket.

Basic Factorisation: $6x + 9 \rightarrow 3(2x + 3)$.
Quadratic Factorisation: $x^2 + 5x + 6 \rightarrow (x + 2)(x + 3)$.
Difference of Two Squares: $a^2 - b^2 = (a - b)(a + b)$.
Grouping: $ax + ay + bx + by \rightarrow a(x + y) + b(x + y) \rightarrow (a + b)(x + y)$.

The laws of indices apply to algebraic terms as they do to numbers.

Multiplication: $x^a \times x^b = x^{a+b}$
Division: $x^a \div x^b = x^{a-b}$
Power of a Power: $(x^a)^b = x^{ab}$
Distribution over multiplication: $(xy)^a = x^a y^a$
Example: Simplify $\frac{12x^5 y^3}{4x^2 y}$.
- $\frac{12}{4} \cdot x^{5-2} \cdot y^{3-1} = 3x^3 y^2$.