Variable

Definition

A variable is a symbol, typically a letter, that represents an unknown or changeable quantity in an algebraic expression or equation. It is a fundamental concept in algebra that allows for the generalization of mathematical relationships and the solution of problems involving unknown values.

5 Must Know Facts For Your Next Test

1. Variables are used to represent unknown or changing values in algebraic expressions and equations.
2. The value of a variable can be determined by solving an equation or evaluating an expression.
3. Variables are essential for translating real-world problems into mathematical form and finding solutions.
4. The properties of equality, such as the addition and subtraction properties, are used to solve equations involving variables.
5. Variables can have different types, such as integers, fractions, or decimals, and their values can be manipulated using various algebraic operations.

Review Questions

• Explain how variables are used in the context of evaluating, simplifying, and translating algebraic expressions.
  • Variables are the building blocks of algebraic expressions, allowing for the representation of unknown or changing values. When evaluating an expression, the variable is replaced with a specific value, and the expression is then calculated. Simplifying an expression involves using the properties of operations, such as the distributive property, to reduce the expression to its simplest form. Translating a real-world problem into an algebraic expression involves identifying the relevant variables and representing the relationships between them using mathematical symbols and operations.
• Describe the role of variables in solving equations using the addition, subtraction, division, and multiplication properties of equality.
  • Variables are essential in solving equations, as they represent the unknown quantities that need to be determined. The properties of equality, such as the addition, subtraction, division, and multiplication properties, are used to isolate the variable and find its value. For example, when solving an equation like $2x + 5 = 17$, the addition and subtraction properties can be used to isolate the variable $x$ by subtracting 5 from both sides and then dividing both sides by 2. This process allows the variable to be solved for, revealing its value in the context of the equation.
• Analyze how variables are used in the context of solving polynomial expressions, including operations such as addition, subtraction, multiplication, and division.
  • Variables play a crucial role in the manipulation and simplification of polynomial expressions. When adding or subtracting polynomials, the like terms with the same variable and exponent are combined. In multiplication, the variables are multiplied according to the product rule, where the exponents are added. Division of monomials involves dividing the coefficients and subtracting the exponents of the variables. The introduction of variables in polynomial expressions allows for the generalization of mathematical relationships and the solution of more complex problems. Understanding the behavior of variables within these operations is essential for working with and solving polynomial expressions.

Related terms

Constant: A constant is a fixed value that does not change within a given context or problem.

Expression: An expression is a mathematical phrase that combines variables, constants, and operations to represent a value or relationship.

Equation: An equation is a mathematical statement that expresses the equality of two expressions, typically involving one or more variables.