Problem 79 - College Algebra

Question

State the name of the property illustrated. \((2+3)+(4+5)-(4+5)+(2+3)\)

Step-by-Step Solution

Answer

The property illustrated is the Commutative property of addition.

Step 1: Recognize the Property

The initial step involves recognizing the property being demonstrated. Look at the provided statement \((2+3)+(4+5)-(4+5)+(2+3)\). Here, the terms are just rearrangements of one another, so this suggests the Commutative property of addition is being used. To confirm this, let's break down the expression.

Step 2: Breaking Down the Expression

Breakdown the expression into simpler terms. \((2+3)+(4+5)-(4+5)+(2+3)\) is the same as \(5 + 9 - 9 + 5\). As we see, additions and subtractions are happening in different order but resulting the same value. This confirms Commutative property of addition.

Step 3: State the Property

After breaking down the expression and verifying that changing the order of terms doesn't affect the result, it can finally be stated that the property illustrated is the 'Commutative property of addition'.

Key Concepts

Properties of Addition

When working with numbers, understanding the Properties of Addition can help simplify and solve problems with ease. One important property is the Commutative Property of Addition. This property states that when two numbers are added, the order does not affect their sum.

In simpler terms, for any numbers a and b, the equation \(a + b = b + a\) holds true. This helps in rearranging and simplifying expressions without changing their value.

For example, if you have \((2 + 3) + (4 + 5) - (4 + 5) + (2 + 3)\), even though the numbers are rearranged, the outcome remains the same due to the commutative property. This makes calculations more flexible and understanding this property can boost your confidence in algebraic manipulations.

College Algebra Problems

College Algebra often involves complex expressions where recognizing patterns, such as the Commutative Property, is key. These properties help simplify expressions, making it essential in solving problems efficiently.

When tackling algebra problems, remember that:

• Recognizing properties saves time and effort.
• Properties like commutative, associative, and distributive can simplify complex expressions.
• Checking work using these properties can confirm the accuracy of solutions.

By mastering these techniques, students can focus on solving equations without getting bogged down by arithmetic complexities.

Mathematical Expressions

Mathematical expressions can often appear daunting. However, properties of addition provide a toolkit for dealing with them efficiently. A mathematical expression can include numbers, variables, and operators like addition and subtraction, all working together to represent a value.

For expressions such as \((2 + 3) + (4 + 5) - (4 + 5) + (2 + 3)\), identifying relationships or properties, like the commutative nature of addition, allows us to rearrange and simplify.

This effectively transforms complex-looking problems into manageable tasks. By breaking expressions down into parts and understanding how properties apply, students can gain clarity and solve problems more systematically.