Problem 13 - Algebra and Trigonometry

Question

State the property of real numbers being used. \((x+2 y)+3 z=x+(2 y+3 z)\)

Step-by-Step Solution

Answer

Associative property of addition.

Step 1: Identify the Operation

Examine the expression to see that it involves addition, specifically the addition of three terms: \((x + 2y) + 3z\).

Step 2: Analyze the Rearrangement

Notice that the terms are being grouped differently: \((x + 2y) + 3z\) is rearranged to \(x + (2y + 3z)\).

Step 3: Recognize the Associative Property

Understand that the associative property of addition allows us to regroup terms: \((a + b) + c = a + (b + c)\). This means the way in which the numbers are grouped does not affect the sum.

Key Concepts

Real Numbers

Real numbers are the numbers you encounter in everyday life, from counting objects to measuring distances. They include both rational numbers (like fractions and integers) and irrational numbers (like the square root of two or \(\pi\)).
Real numbers can be positive, negative, or zero, and are represented on the number line.

Their versatility makes them a fundamental part of mathematics. When dealing with real numbers, certain properties like commutativity, associativity, and distributivity often come into play.

• Rational Numbers: These can be expressed as a fraction \(\frac{p}{q}\), where \q\ is not zero.
• Irrational Numbers: These cannot be written as simple fractions; their decimal expansions are non-repeating and non-terminating.

Understanding real numbers helps in solving a variety of mathematical problems, including those that use properties of addition.

Addition of Terms

The addition of terms involves combining numbers or algebraic expressions to find their sum. Regardless of how complicated they look, the process of addition is built on simple foundations.
These foundations include the syntax of writing terms and using operators like the plus sign (+) to indicate addition.

In the expression \(x + 2y + 3z\), each part (\(x, 2y,\) or \(3z\)) is a term that can be added together. Consider these basic principles:

• Collect like terms: Terms that share the same variable part can be combined by adding the coefficients.
• Ensure proper parentheses: These define which operations to perform first due to the associative property.

Understanding how terms add up is crucial for simplifying and solving algebraic expressions.

Rearrangement of Expressions

Rearranging expressions in mathematics is a key part of simplifying and solving equations. It allows you to organize terms in a way that can reveal meaningful patterns or simplifications.
The expression \( (x + 2y) + 3z \) is rearranged to \( x + (2y + 3z) \). This kind of rearrangement does not change the sum due to properties like associativity.

Here are some typical steps in rearranging expressions:

• Identify groupings: Look at how terms are grouped in parentheses.
• Apply properties: Use properties such as commutativity and associativity to shift and regroup terms.

This manipulation can set the stage for easier computation or further transformations in problem-solving.

Properties of Addition

The properties of addition make our calculations precise and predictable. They are essential for understanding and executing algebraic manipulations.
Let's look at the most important properties related to addition in the context of real numbers:

• Commutative Property: This property states that you can change the order of the addends. For example, \( a + b = b + a \).
• Associative Property: This allows for regrouping of addends without changing their sum, as seen in \( (a + b) + c = a + (b + c) \). This was exactly what was used in the given expression.
• Identity Property: Adding zero to any number doesn’t change its value, as \( a + 0 = a \).

When using these properties, expressions become more flexible and manageable, which is key to mastering algebra.