Algebraic Expressions

Definition

An algebraic expression is a mathematical phrase that combines variables, numbers, and operations such as addition, subtraction, multiplication, and division. These expressions represent unknown quantities and allow for the manipulation and analysis of relationships between different quantities.

5 Must Know Facts For Your Next Test

1. Algebraic expressions are used to solve equations, model real-world situations, and represent relationships between quantities.
2. The properties of equality, such as the addition and subtraction properties, are used to solve equations involving algebraic expressions.
3. When adding and subtracting integers within an algebraic expression, the rules for integer operations must be applied.
4. Algebraic expressions with fractions can be solved by applying the properties of fractions, such as finding common denominators.
5. The commutative, associative, and distributive properties of operations can be used to simplify and manipulate algebraic expressions.

Review Questions

• Explain how the addition and subtraction properties of equality can be used to solve equations involving algebraic expressions.
  • The addition and subtraction properties of equality state that the same quantity can be added to or subtracted from both sides of an equation without changing the solution. When solving equations with algebraic expressions, these properties can be used to isolate the variable term on one side of the equation, allowing you to find the value of the unknown. For example, to solve the equation $2x + 5 = 11$, you would subtract 5 from both sides to isolate the $x$ term, resulting in $2x = 6$, and then divide both sides by 2 to find $x = 3$.
• Describe the process of adding integers within an algebraic expression and how it relates to the properties of integer operations.
  • When adding integers within an algebraic expression, you must apply the rules for integer operations. If the integers have the same sign, you add their absolute values and keep the common sign. If the integers have different signs, you subtract the smaller absolute value from the larger absolute value and keep the sign of the larger integer. For example, in the expression $3x - 2y + 4x - 5y$, the addition of the $x$ terms ($3x + 4x = 7x$) and the subtraction of the $y$ terms ($-2y - 5y = -7y$) would result in the simplified expression $7x - 7y$.
• Analyze how the commutative, associative, and distributive properties can be used to manipulate and simplify algebraic expressions.
  • The commutative property states that the order of the terms in an addition or multiplication expression does not affect the result. The associative property allows for the grouping of terms in an addition or multiplication expression without changing the outcome. The distributive property states that multiplication can be distributed over addition or subtraction, allowing for the simplification of expressions with multiple operations. By applying these properties, you can rearrange, group, and combine terms within an algebraic expression to obtain a simplified, equivalent form. For example, using the distributive property, the expression $3(2x + 5)$ can be simplified to $6x + 15$.

Related terms

Variable: A symbol, usually a letter, that represents an unknown or changeable quantity in an algebraic expression.

Coefficient: The numerical factor multiplied by a variable in an algebraic expression.

Constant: A fixed value in an algebraic expression that does not change.