Algebraic Language

Algebraic Language

Basic Terminology

  • Constants: Fixed values that do not change, such as 5, -3, or π.
  • Variables: Letters used to represent numbers, such as x, y, or z.
  • Coefficients: Numbers multiplying variables, such as the 3 in 3x.
  • Expressions: Combinations of constants, variables, coefficients, and operations, such as 3x - 2.


  • Addition (+) and Subtraction (-): Combine or remove quantities. For example, x + 2 or 5 - x.
  • Multiplication (x) and Division (÷): Increase or reduce quantities. For example, 2x or x ÷ 3.
  • Exponentiation (^): Raise a number to a power. For example, x^2 or 2^x.
  • Root (√): Extract the root of a number. For example, √x or 4√x.

Algebraic Terms and Statements

  • Term: A single part of an algebraic expression, separated by plus or minus signs. For example, in the expression 3x + 2y - 7, the terms are 3x, 2y and -7.
  • Equation: A mathematical statement that establishes the equality of two expressions. For example, 3x - 2 = 4.
  • Identity: An equation that is true for all possible values of the variable. For example, (a + b)^2 = a^2 + 2ab + b^2.
  • Inequality: A statement that one expression may be greater than (<), less than (>), or not equal to (≠) another. For example, x > 3.
  • Function: An expression that produces a single output (y) for every input (x). For example, y = 2x + 3.

Manipulating and Simplifying Expressions

  • Like Terms: Terms that contain the same variables raised to the same power. Like terms can be added or subtracted. For example, in 4x + 3x, the like terms are 4x and 3x.
  • Solving Equations: The process of finding the values of the variable that make the equation true. Utilise the properties of equality.
  • Factoring: The process of writing an expression as the product of its factors. For example, 3x^2 - 12x = 3x(x - 4).
  • Expanding: Apply the distributive law to remove parentheses. For example, 2(x + 3) = 2x + 6.

Useful Techniques

  • Substitution: Replace variables with known or given values to solve equations or evaluate expressions.
  • Rearranging Equations: Manipulate equations to isolate desired variables, often using operations in reverse.
  • Completing the Square: A technique used to solve quadratic equations, write equations in vertex form, or graph parabolas.
  • Quadratic Formula: A method to solve any quadratic equation, found by completing the square on ax^2 + bx + c = 0.