Problem 9 - Intermediate Algebra

Question

State the property that justifies each of the statements. For example, \(3+(-4)=(-4)+3\) because of the commutative property of addition. $$ 12 y x=12 x y $$

Step-by-Step Solution

Answer

The commutative property of multiplication.

Step 1: Identify Operation

The given expression is an equation involving multiplication: \( 12yx = 12xy \).

Step 2: State the Property

The statement \( 12yx = 12xy \) is an example of the commutative property of multiplication.

Key Concepts

Commutative Property of Multiplication

The commutative property of multiplication is an essential principle in mathematics. It states that the order in which you multiply numbers does not affect the product. In other words, if you have two numbers, say \( a \) and \( b \), then \( a \times b = b \times a \). This property is very helpful because it allows flexibility when solving problems, which can make calculations easier and often more intuitive.
For example, when you are given an expression like \( 3 \times 4 \) or \( 4 \times 3 \), using the commutative property, you know these are equal because both result in the same product: 12.
This property can be particularly useful in algebra as it often simplifies solving equations or expressions by rearranging terms to group like factors or terms together.

Properties of Operations

Mathematics involves various properties of operations that help in simplifying expressions and performing calculations. These properties are rules that numbers follow and include the commutative, associative, identity, and distributive properties:

• **Commutative Property:** As discussed, it applies to both addition and multiplication and states that changing the order does not change the result.
• **Associative Property:** This property states that when you add or multiply, the way you group the numbers does not change the sum or product. For instance, \( (2+3)+4 = 2+(3+4) \).
• **Identity Property:** This property states that adding 0 to a number leaves it unchanged (additive identity), and multiplying a number by 1 leaves it unchanged (multiplicative identity).
• **Distributive Property:** This property connects addition and multiplication, stating that \( a(b+c) = ab + ac \).

These properties are fundamental as they form the basis of many mathematical rules and problems, thereby assisting in performing more complex calculations with ease.

Multiplication in Algebra

In algebra, multiplication goes beyond the basic operation learned in arithmetic. It involves working with variables, coefficients, and sometimes even exponents. Understanding how multiplication interacts with variables is central to solving algebraic expressions and equations.
In expressions like \( 3ab \), multiplication implies that 3 is multiplied by both \( a \) and \( b \). When variables are involved, properties of operations, specifically the commutative and associative properties, are often utilized for simplification and manipulation.
For example, rearranging \( 5xy \) to \( yx \times 5 \) is valid under the commutative property, as multiplication can occur in any order without changing the outcome. This is especially useful when needing to factor expressions, solve for a specific variable, or combine like terms.
Mastering multiplication in algebra not only aids in problem-solving but also provides the foundation for more advanced mathematical concepts you will encounter as you progress.