Problem 18 - Algebra and Trigonometry

Question

State the property of real numbers being used. \(7(a+b+c)=7(a+b)+7 c\)

Step-by-Step Solution

Answer

Distributive Property of Multiplication over Addition.

Step 1: Identify the Distributive Property

The given equation is of the form where a single term is distributed across terms within a parenthesis. This form typically represents the distributive property of multiplication over addition for real numbers. In this case, the term 7 is being distributed to each term inside the parenthesis.

Step 2: Apply the Distributive Property

According to the distributive property, multiplying a number by a sum is the same as multiplying the number by each addend separately and adding the results. In mathematical terms: \[ a(b+c) = ab + ac \]Applying this to the given expression:\[ 7(a+b+c) = 7(a+b) + 7c \]

Key Concepts

Properties of Real Numbers

Real numbers encompass a broad set that includes rational and irrational numbers such as integers, fractions, and decimals. These numbers follow specific properties that help simplify mathematical expressions and solve equations efficiently. Some properties include:

• Commutative Property: Order does not affect the result. For addition, \( a + b = b + a \), and for multiplication, \( ab = ba \).
• Associative Property: Grouping of numbers does not affect the sum or product. For example, \( (a + b) + c = a + (b + c) \) and \( (ab)c = a(bc) \).
• Identity Property: Adding zero or multiplying by one leaves the number unchanged. \( a + 0 = a \) and \( a \cdot 1 = a \).
• Distributive Property: Multiplication distributed over addition. \( a(b+c) = ab + ac \).

These fundamental properties form the basis for algebraic manipulation and are essential for solving equations involving real numbers.

Algebraic Expressions

Algebraic expressions are mathematical phrases combining numbers, variables, and operators. They represent quantities in a general form, using algebra to model real-world situations. For instance, the expression \( 7(a+b+c) \) is an example.Breaking down algebraic expressions involves:

• Constants: Fixed values like 7 in \( 7(a+b+c) \).
• Variables: Symbols like \( a, b, \text{and} c \) representing unknown values or numbers that can change.
• Operators: Symbols like \(+, \times, -\) that define operations to be performed, such as addition and multiplication.

Algebraic expressions allow us to simplify, solve, and model equations. Recognizing how elements combine under properties like distribution is crucial to understanding their behavior and simplifying complex formulas.

Multiplication Over Addition

The process of multiplying a number across a sum is known as multiplication over addition, emphasized through the distributive property. Let's delve deeper into how this works.When we encounter a scenario like \( 7(a+b+c) \), the distributive property comes into play. Here's how it works:

• Each term inside the parentheses is individually multiplied by 7.
• As per the example given, \( 7 imes (a+b+c) \) simplifies through distribution: \( 7a + 7b + 7c \).
• This operation allows us to break down complex expressions into simpler parts without altering the overall value.

Such breakdowns are beneficial in moving seamlessly through algebraic equations. This efficient handling of expressions ensures a smoother pathway from problem to solution.