Problem 23 - Prealgebra

Question

Apply the associative property to expression, and then simplify the result. \((6 y+4)+3\)

Step-by-Step Solution

Answer

The simplified expression is \(6y + 7\).

Step 1: Identify the Expression

The given expression is \((6y + 4) + 3\). The task is to apply the associative property.

Step 2: Apply the Associative Property

The associative property states that changing the grouping of numbers in addition or multiplication does not change the result. Therefore, we can rewrite the expression as \(6y + (4 + 3)\).

Step 3: Simplify the Expression

We now simplify the expression inside the parentheses: \(4 + 3 = 7\). Substitute this back in to get \(6y + 7\).

Step 4: Result Verification

The expression is now simplified to \(6y + 7\). Ensure that all rules and calculations are correctly applied with the associative property.

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra that involves combining like terms and making the expression easier to work with. When you are given an expression like \((6y + 4) + 3\), the goal is to break it down into its simplest form so that it is easier to understand and manipulate in further calculations. When simplifying, you might:

• Combine numbers without variables.
• Group like terms together.
• Use properties such as the associative or commutative properties to rearrange terms when necessary.

By simplifying \((6y + 4) + 3\) to \(6y + 7\), it becomes a more straightforward expression that can be easily used in equations, inequalities, and other mathematical operations. It's important to practice simplification regularly, as it's a foundation for more advanced topics in algebra.

Prealgebra Concepts

Prealgebra introduces foundational concepts that prepare students for algebra. One of the key ideas is understanding mathematical properties, such as the associative property used in grouping terms. In expressions like \((6y + 4) + 3\), prealgebraic thinking helps determine how components can be reassessed or reordered to facilitate ease in calculations.
In this context, engaging with basic operations—addition, subtraction, multiplication, division—and handling simple algebraic expressions set the stage for algebraic reasoning. Skills developed in prealgebra are crucial and include:

• Mastering number operations and basic manipulation of expressions.
• Gaining familiarity with properties of operations (associative, commutative, and distributive).
• Understanding variables and how they can be used to represent numbers.

Prealgebra is essentially about building confidence and competence in handling numerical and algebraic expressions, paving the way for more complex algebraic work.

Algebraic Expressions

An algebraic expression consists of variables, constants, and operation symbols like \(+\), \(-\), and \(\times\). In our example, \((6y + 4) + 3\), we deal with the algebraic term \(6y\) and constant terms \(4\) and \(3\). Understanding how to manipulate these components is essential in problem-solving.
Algebraic expressions are often used to model real-world situations, and learning how to simplify or alter these expressions is key to finding solutions. Here are some tips when working with algebraic expressions:

• Identify and separate different types of terms: constants and variable terms.
• Apply properties such as the associative property to group terms for easier calculations.
• Simplify expressions by combining like terms or reducing their complexity.

Efficient handling of algebraic expressions can significantly simplify otherwise complex problems, making them more manageable and understandable. This lays the groundwork for solving equations and systems of equations in algebra.