Introduction to algebra Revision Notes

In algebra, letters such as x, y, or a are used to represent numbers whose values can change or are not yet known. These letters are called variables. Using variables allows us to write general expressions and formulas that apply to many different numbers, rather than just one specific number.

For example, in the expression 5x + 2, the letter x stands for any number. This means the expression can represent many different values depending on what number x takes. Variables make algebra a powerful tool for solving problems involving unknown or changing quantities.

• Variables are symbols representing general numbers.
• They allow expressions to be general rather than specific.
• Common variable letters include x, y, a, b, and n.

An algebraic expression combines numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. For example, 3a - 4, 2x + 7, and 5y² are all algebraic expressions.

Each expression can be evaluated by substituting a specific number for the variable(s). This process turns the expression into a numerical calculation.

• Expressions include numbers and variables combined by operations.
• They can be simple (like 4x) or more complex (like 2x + 3y - 5).
• Evaluating expressions requires substitution of values for variables.

Substitution means replacing the variable in an expression with a given numerical value. After substitution, the expression becomes a calculation with numbers only, which can be evaluated to find a result.

For example, if the expression is 3a + 4 and a = 2, substitute 2 for a: 3 × 2 + 4 = 6 + 4 = 10.

It is important to substitute all occurrences of the variable and follow the correct order of operations when evaluating.

• Replace every variable with the given number.
• Use brackets when substituting to avoid errors, e.g., 3(a + 2) with a = 4 becomes 3(4 + 2).
• Calculate step-by-step following arithmetic rules.

Example 1: Evaluate 2x + 5 when x = 3. Substitute 3 for x: 2 × 3 + 5 = 6 + 5 = 11.

Example 2: Evaluate 4y - 7 when y = 5. Substitute 5 for y: 4 × 5 - 7 = 20 - 7 = 13.

Example 3: Evaluate 3(a + 2) when a = 4. Substitute 4 for a: 3 × (4 + 2) = 3 × 6 = 18.

• Always substitute carefully and check each step.
• Use brackets to clarify substitution especially when variables are inside parentheses.
• Verify the final answer by re-calculating if unsure.

Variables are usually single letters and can appear multiple times in an expression. Each occurrence must be replaced during substitution.

When substituting negative numbers or expressions, use brackets to avoid sign errors. For example, if x = -2, then 3x means 3 × (-2) = -6.

Remember to follow the order of operations: brackets, powers, multiplication and division, addition and subtraction (BIDMAS).

• Replace all instances of the variable.
• Use brackets when substituting negative or complex values.
• Follow the order of operations carefully to avoid mistakes.