Problem 68 - Algebra 2

Question

PREREQUISITE SKILL. Use the Distributive Property to find each product. $$ 3(x-z) $$

Step-by-Step Solution

Answer

The product is \(3x - 3z\).

Step 1: Identify the Expression

The original expression given is \(3(x-z)\). We need to distribute the number 3 to each term inside the parentheses.

Step 2: Distribute the 3

Apply the distributive property which states that \(a(b+c) = ab + ac\). Distribute the 3 to each term inside the parentheses: \(3 \times x\) and \(3 \times (-z)\).

Step 3: Calculate Each Product

First, calculate \(3 \times x = 3x\). Then, calculate \(3 \times (-z) = -3z\).

Step 4: Combine the Products

Combine the products from Step 3: \(3x + (-3z)\). This simplifies to \(3x - 3z\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They form the building blocks of algebra, used to represent mathematical relationships and solve problems. An expression can be as simple as a single number or variable, or more complex with multiple terms involving operations such as addition and multiplication.
In our given exercise, the expression is \(3(x-z)\). Here, 3 is a constant, while \(x\) and \(z\) are variables representing unknown values. The parentheses indicate grouping, showing us which part of the expression to consider together when performing operations. Understanding how to navigate and manipulate these expressions is crucial in developing strong algebra skills.

Mastering Multiplication in Algebra

Multiplication in algebra involves multiplying constants and variables. It is crucial to correctly apply multiplication rules in expressions to reach accurate solutions. Multiplying constants by variables involves simply placing them beside each other, as shown in distributed terms like \(3x\).
When dealing with expressions involving parentheses, multiplication can extend to all terms within. This is where the Distributive Property comes into play, allowing multiplication to distribute over addition or subtraction within the parentheses. For example, in the expression \(3(x-z)\), the constant 3 multiplies with both \(x\) and \(-z\), demonstrating how multiplication is expanded within algebraic structures.

The Art of Distributing Constants

Distributing constants throughout algebraic expressions is a technique that simplifies expressions and makes them easier to work with. This process follows the Distributive Property, formulated as \(a(b + c) = ab + ac\). It states that a single term multiplied by a sum or difference should evenly multiply with each term inside the parentheses individually.
In the given exercise, \(3(x-z)\) transforms by distributing 3 to each variable: \(3 \times x\) becomes \(3x\) and \(3 \times (-z)\) becomes \(-3z\). This division of work simplifies the initial expression into a straightforward format of \(3x - 3z\). Mastering this property is essential in tackling more complex algebraic problems efficiently.

Building Basic Algebra Skills

Having a solid foundation in basic algebra skills is essential for tackling more advanced mathematical concepts. These skills include simplifying expressions, solving equations, and understanding the properties of numbers and operations.
Using the Distributive Property to expand expressions, like we did in \(3(x-z)\), is a fundamental skill. It helps in simplifying expressions by breaking them down into manageable parts.
Here are some key strategies:

• Practice regularly to reinforce understanding and fluency.
• Work on recognizing patterns within expressions.
• Focus on understanding properties, such as the Distributive Property, to simplify mathematical tasks.

By applying these skills, students can approach algebra with confidence and tackle a broad array of mathematical challenges.