Problem 8 - Algebra for College Students

Question

State the property that justifies each of the statements. For example, \(3+(-4)=(-4)+3\) because of the commutative property of addition. $$-3(2+4)=-3(2)+(-3)(4)$$

Step-by-Step Solution

Answer

The distributive property of multiplication over addition.

Step 1: Identify the Operation

Look at the given statement: \[-3(2 + 4) = -3(2) + (-3)(4)\]This expression involves multiplication of numbers and addition as an operation.

Step 2: Apply the Distributive Property

The equation shows \[-3(a + b) = -3a + -3b\] This matches the Distributive Property of Multiplication over Addition, which states that multiplying a sum by a number is the same as doing each multiplication separately. Thus, \[-3(2 + 4) = -3(2) + (-3)(4)\] demonstrates using the distributive property.

Key Concepts

Multiplication over Addition

The concept of Multiplication over Addition is captured by the Distributive Property. This property is one of the fundamental rules of arithmetic, describing how multiplication interacts with addition. It states that if you have a multiplication operation involving a sum, you can distribute the multiplication to each term in the sum separately. For example, in the expression \[-3(2 + 4)\], we are multiplying \(-3\) by the sum of \(2\) and \(4\). According to the Distributive Property, this is equivalent to multiplying \(-3\) by \(2\) and \(-3\) by \(4\) separately, and then adding the results: \[-3 \times 2 + -3 \times 4\].

• Helps simplify expressions.
• Makes calculations manageable.
• Applies to variables and real numbers alike.

This principle is incredibly useful in algebra, as it allows for the manipulation and simplification of expressions, which is a common task in solving equations.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. In the given exercise, the expression \(-3(2 + 4)\) is an example of using numbers without variables. However, the same principle applies when variables are included. For example, consider the expression \(-3(x + y)\), where \(x\) and \(y\) are variables. Using the Distributive Property, this becomes \(-3x + -3y\), which can be further simplified if \(x\) and \(y\) have known values.Algebraic expressions are the building blocks of equations and functions. They help represent real-world situations in mathematical terms and are a crucial part of solving algebra problems. Here’s what makes them important:

• They provide a way to generalize arithmetic operations with unknowns.


• Allow for the representation of complex relationships in simple terms.


• Facilitate problem-solving and logical reasoning.

Properties of Real Numbers

The Properties of Real Numbers are a set of rules that apply to all real numbers, which include positive and negative numbers, fractions, and decimals. These properties, such as commutative, associative, identity, and the distributive property, help us understand how numbers behave under different operations. In the provided exercise, we use the Distributive Property, which specifically relates to how multiplication interacts with addition. This property ensures that calculations can be reordered and simplified without changing the result.
The properties of real numbers help in:

• Ensuring consistency in mathematical operations.
• Providing foundational rules that guide calculations.
• Building a framework for understanding more complex mathematical concepts.

Understanding these properties is crucial for solving algebraic problems efficiently and accurately. They are fundamental to both simple arithmetic and more advanced mathematics.