Problem 46 - Algebra A Combined Function

Question

Use the distributive property to write each expression without parentheses. Then simplify the result, if possible. See Examples 7 through 12. $$ 8(3 y+z-6) $$

Step-by-Step Solution

Answer

Final expression: \(24y + 8z - 48\).

Step 1: Apply Distributive Property

The distributive property states that \(a(b+c) = ab + ac\). We will apply this property to distribute the 8 across all terms in the parentheses: \(8(3y + z - 6)\). This gives us \(8 \times 3y + 8 \times z + 8 \times (-6)\).

Step 2: Perform Multiplication

Now, calculate each of the individual products: \(8 \times 3y = 24y\), \(8 \times z = 8z\), and \(8 \times (-6) = -48\).

Step 3: Combine the Terms

Combine all the products from the multiplication: \(24y + 8z - 48\).

Step 4: Simplify the Expression

In this particular expression, there are no like terms to combine further. So, the expression is already simplified.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. They form the foundation of algebra and are used to express mathematical ideas concisely. When analyzing algebraic expressions, it's crucial to identify all the components:

• Constants: These are the numbers on their own, like -6 in the provided example.
• Variables: Letters that represent unknown values, such as y and z in the example.
• Coefficients: Numbers that are multiplied by variables, like 8 (multiplier) and 3 (coefficient of y) in the distributive property step.

The goal is often to manipulate these expressions using different mathematical techniques to simplify, evaluate, or solve them based on a problem's requirements. Understanding how each part functions within an expression is fundamental to proceeding with operations like distribution.

Simplification of Algebraic Expressions

Simplification involves reducing an expression to its most concise and manageable form. When simplifying, the objective is to combine like terms and perform operations that make the expression as straightforward as possible.For example, in the expression obtained from using distributive property: \(24y + 8z - 48\), simplification checks:- **Combining Like Terms**: Look for terms with the same variable raised to the same power. Here, there are no terms with similar variables, so no further combination is possible.- **Arithmetic Simplification**: Perform basic arithmetic operations wherever possible. For instance, the original multiplication step "simplifies" into terms we clearly understand.Simplification is key for solving algebraic equations, as it transforms complex expressions into solvable ones. When expressions are too simplified, they might stay unchanged, like in our example, indicating the process is complete.

Insight into Mathematical Properties

Mathematical properties are the rules that govern the operations within an expression and formulae. They are essential for understanding and manipulating algebraic expressions effectively.One of the most important properties involved in simplifying expressions is the **Distributive Property**:- **Distributive Property**: This property states that \(a(b+c) = ab + ac\). It allows us to spread a single multiplying factor across a sum or difference inside parentheses. In the example \(8(3y + z - 6)\), the 8 multiplies each term individually, resulting in \(24y + 8z - 48\).Other critical mathematical properties include:- **Commutative Property**: Which deals with the order of operations for addition and multiplication.- **Associative Property**: Which focuses on how numbers are grouped in operations.Utilizing these properties not only simplifies algebraic expressions but also helps in solving complex equations effectively. Understanding how and when to apply these properties is a crucial skill in mastering algebra.