Algebraic Expressions

Algebra is a language for describing patterns before you know the exact numbers. A letter such as $x$ or $n$ lets you write one rule that works for many possible values.

CSEC tests algebra at all three cognitive levels: knowing symbols, simplifying expressions correctly, and using expressions to model word problems. When you see an example, focus on what each symbol represents and why each operation is being used. That habit makes later topics like equations, functions, and formulae much easier.

Symbolic Representation¶

Symbols let you describe a general situation. Before simplifying or solving, decide what each letter stands for; otherwise the expression becomes a string of signs instead of a statement with meaning.

In algebra, we use letters to represent numbers:

• $x$, $y$, $a$, $b$ = variables (unknown quantities)
• $3x$ = 3 times some number
• $x + 5$ = some number plus 5
• $2a - b$ = twice $a$, minus $b$

Translating Between Words and Algebra¶

Word problems become easier when you translate one phrase at a time. Look for operation words: "sum" suggests addition, "difference" suggests subtraction, "product" suggests multiplication, and "quotient" suggests division.

Directed Numbers (Signed Numbers)¶

Directed numbers show direction from zero. A negative sign may mean debt, below sea level, loss, reverse movement, or temperature below zero, depending on the context.

Positive numbers are greater than zero. Negative numbers are less than zero.

$(-3) + (+5) = 2$ $(-3) + (-5) = -8$ $(-3) \times (-5) = 15$

Rules:

• Same signs multiply to positive: $(+) \times (+) = (+)$ and $(-) \times (-) = (+)$
• Different signs multiply to negative: $(+) \times (-) = (-)$

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Part 2: Operations with Algebraic Expressions¶

Adding and Subtracting Like Terms¶

Like terms can be combined because they count the same kind of object. Three $x$'s plus five $x$'s make eight $x$'s, but three $x$'s plus five $y$'s must stay separate.

Like terms have the same variable and power.

$3x + 5x = 8x$ $2a^2 + 3a^2 = 5a^2$ $3x + 5y \text{ cannot be combined (different variables)}$

Multiplication and Division of Expressions¶

Multiplication spreads a factor across every term in a bracket. Division does the reverse by sharing each term by the same divisor.

Multiply: Use the distributive law $a(b + c) = ab + ac$

$2 \times (x + 3) = 2x + 6$ $x \times (x + 5) = x^2 + 5x$

Divide: Divide each term by the divisor

$\frac{6x + 9}{3} = \frac{6x}{3} + \frac{9}{3} = 2x + 3$

Expanding Brackets¶

Expanding removes brackets by multiplying. It is useful when an expression must be simplified or compared with another expression in standard form.

Distributive law: $a(b + c) = ab + ac$

Double brackets: $(a + b)(c + d) = ac + ad + bc + bd$

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Part 3: Substitution and Binary Operations¶

Substitution — Putting Numbers Into Expressions¶

Substitution turns a general expression into a specific value. Brackets are important because they preserve signs, especially when substituting negative numbers.

Substitution means: replace the letter with the number you're given, then calculate.

Why this matters: Many algebra problems ask "What happens if x equals this number?" Substitution lets you find the answer by plugging in the value.

The Three-Step Process¶

Step 1: Identify which variable is being replaced and what value it gets

Step 2: Replace every instance of that letter with the value (use brackets to avoid confusion)

Step 3: Follow the order of operations (PEMDAS: Powers, Multiplication/Division, Addition/Subtraction)

Example 1: Simple Expression¶

Example 2: Multiple Variables¶

When Substitution Goes Wrong¶

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Binary Operations — Custom Number Rules¶

For binary operations, the symbol is only a label for a rule. Do not assume it behaves like ordinary multiplication or addition unless the definition says so.

A binary operation is a special rule that tells you how to combine two numbers. It's not just the normal operations (+, −, ×, ÷) — it can be anything!

Why learn this? Tests use binary operations to check if you can follow instructions and substitute correctly. It's a skill test, not a math test.

Reading a Binary Operation Definition¶

When you see "$a * b = 2a + b$", it means:

• $*$ is the special symbol (could be anything: ★, ⊕, ◆, etc.)
• $a$ is the FIRST number
• $b$ is the SECOND number
• The rule is: multiply the first number by 2, then add the second number

Nested Operations (Operations Inside Operations)¶

When you see $2 * (3 * 1)$, solve the parentheses FIRST.