Problem 63 - Algebra 2

Question

Name the property illustrated by each equation. \(2 x+4 y+3 z=2 x+3 z+4 y\)

Step-by-Step Solution

Answer

The equation illustrates the Commutative Property of Addition.

Step 1: Identify the Equation Type

The given equation is \(2x + 4y + 3z = 2x + 3z + 4y\). This equation consists of expressions containing the same terms rearranged in different orders.

Step 2: Recognize the Property Used

The property of addition that allows terms to be rearranged without changing the sum is called the Commutative Property of Addition. This property shows that no matter how the terms are ordered, the result is still the same.

Key Concepts

Properties of Addition

The Properties of Addition are foundational rules that apply to operations involving addition. One of the most well-known properties is the **Commutative Property of Addition**. This property states that changing the order of the addends does not affect the sum. For example, in the expression \(a + b = b + a\), the result remains the same whether you add \(a\) to \(b\) or \(b\) to \(a\).
Another key property is the **Associative Property of Addition**, which allows grouping of terms in any order without changing the sum. For instance, \((a + b) + c = a + (b + c)\). These properties simplify complex calculations and help in rearranging equations to find desired outcomes without altering the values.

Equation Solving

Equation solving is a mathematical process of finding the value of variables that satisfy the equation. It requires understanding and applying properties of mathematics, such as the properties of addition, to manipulate and transform equations.
When solving equations, it is crucial to maintain balance. This means performing the same operation on both sides of the equation to keep them equal. This principle is used in rearranging terms and simplifying expressions.

• Start by identifying like terms and constants.
• Apply properties of addition, such as the commutative and associative properties, to simplify expressions.
• Use operations to isolate the variable, maintaining the equation's balance.

This method is vital in solving algebraic equations efficiently.

Order of Operations

The Order of Operations is a set of rules that determines the sequence in which calculations are performed. These rules ensure accurate results by providing a structured way to solve mathematical expressions.
*PEMDAS* (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)) is a well-known acronym that aids in remembering this sequence. First, solve expressions inside parentheses, followed by exponents. Then perform multiplication or division as they appear from left to right, and finally, handle addition or subtraction in sequence.

• Order ensures consistent results across various computations.
• Prevents errors and misinterpretations of complex expressions.
• Critical in understanding and applying mathematical properties during problem-solving tasks.

The order of operations is a fundamental concept in mathematics that organizes and simplifies problem-solving processes.