Identities

Definition

Identities are mathematical expressions that are always true, regardless of the values assigned to the variables. They represent fundamental relationships between mathematical operations and quantities that hold true in all cases.

5 Must Know Facts For Your Next Test

1. Identities are used to simplify and manipulate algebraic expressions, which is crucial for solving linear equations.
2. Common identities include the distributive property, the commutative property, and the associative property, among others.
3. Identities can be used to rewrite expressions in different forms, which can make them easier to work with when solving equations.
4. Recognizing and applying identities is an important skill for solving linear equations using a general strategy.
5. Identities are fundamental to the study of algebra and are used extensively in more advanced mathematical topics.

Review Questions

• Explain how identities can be used to simplify algebraic expressions when solving linear equations.
  • Identities represent fundamental relationships between mathematical operations and quantities that always hold true. When solving linear equations, identities can be used to rewrite and simplify algebraic expressions, making them easier to manipulate and solve. For example, the distributive property, $a(b + c) = ab + ac$, can be used to expand or factor expressions, which is often necessary when solving for the unknown variable in a linear equation.
• Describe how the commutative and associative properties, which are examples of identities, can be applied when solving linear equations.
  • The commutative property, $a + b = b + a$, and the associative property, $(a + b) + c = a + (b + c)$, are identities that can be used to rearrange the terms in a linear equation without changing its solution. This can be helpful when trying to isolate the variable or simplify the equation. For instance, if the equation is $3x + 5 = 7x - 2$, the commutative property can be used to rewrite it as $5 + 3x = 7x - 2$, making it easier to solve for $x$.
• Analyze how identities can be used to verify the equivalence of different algebraic expressions when solving linear equations.
  • Identities can be used to demonstrate that two algebraic expressions are equivalent, which is important when verifying the steps in solving a linear equation. For example, if you have the expression $2(3x + 4) - 5x$ and want to show that it is equivalent to $6x + 8 - 5x$, you can use the distributive property identity, $a(b + c) = ab + ac$, to simplify the first expression and compare it to the second. This process of manipulating expressions using identities can help you ensure that each step in solving a linear equation is valid and leads to the correct solution.

Related terms

Algebraic Expressions: Algebraic expressions are mathematical phrases that combine variables, numbers, and operations to represent a value or relationship.

Equations: Equations are mathematical statements that express the equality of two algebraic expressions.

Equivalent Expressions: Equivalent expressions are algebraic expressions that have the same value, regardless of the values assigned to the variables.