Problem 5 - Algebra and Trigonometry

Question

\(3-10=\) State the property of real numbers being used. $$ (x+2 y)+3 z=x+(2 y+3 z) $$

Step-by-Step Solution

Answer

-7; Associative Property of Addition.

Step 1: Evaluate the Expression

Calculate the result of the given expression: \(3 - 10\). First, identify the operation which is subtraction. Then, subtract 10 from 3: \(3 - 10 = -7\).

Step 2: Identify the Property Used

The given statement \((x + 2y) + 3z = x + (2y + 3z)\) uses a property of real numbers. Notice how the expression has been rearranged without changing its value, this indicates the Associative Property of Addition. This property states \( (a + b) + c = a + (b + c)\).

Key Concepts

Properties of Real Numbers

The properties of real numbers are fundamental rules that dictate how numbers interact in mathematical operations. These rules include important properties such as the commutative, associative, distributive, identity, and inverse properties, among others. Each property simplifies complex algebraic expressions and aids in solving them efficiently.

Understanding these properties allows you to recognize patterns and predict outcomes in mathematical operations. By applying these rules, particularly the associative property in this context, you can rearrange and regroup numbers or variables to make calculations more straightforward. This flexibility is essential in both basic mathematics and more advanced algebraic problems.

Associative Property of Addition

The associative property of addition is one of the most essential properties of real numbers. It means that when you are adding three or more numbers, the way in which you group them doesn't affect the sum. For addition, this can be represented as:

• \((a + b) + c = a + (b + c)\)

This property is called "associative" because it associates or groups various numbers differently while still achieving the same total. It’s highly useful when dealing with large sums or algebraic expressions as it simplifies calculations and can reduce error.

In the exercise, the expression \((x + 2y) + 3z = x + (2y + 3z)\) shows how the associative property allows changing the grouping of \(x, 2y,\) and \(3z\) without altering the value, making it easier to reorder calculations in a convenient manner. This flexibility is crucial for solving equations efficiently.

Addition

Addition is a basic mathematical operation that combines two or more numbers to produce a sum. It is one of the four fundamental operations of arithmetic, alongside subtraction, multiplication, and division. Adding numbers involves computing their total together, regardless of order or grouping.

The beauty of addition, especially when combined with properties like the associative property, is its flexibility. Whether numbers are arranged in different orders or grouped in varied ways, the sum remains consistent. This allows for strategic planning and execution in mathematical operations, making problem-solving more intuitive and less prone to mistakes.

For example, if you have the expression \(a + b + c\), you can decide to add \(a\) and \(b\) first and then add \(c\), or start with adding \(b\) and \(c\) before adding \(a\). Either way, the sum is unaffected, showcasing addition's elegant nature and utility in both simple and complex mathematical scenarios.