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Chapter 1: Problem 37

Determine which of the fundamental laws of algebra is demonstrated. $$6(7)=7(6)$$

Short Answer

Expert verified
The equation demonstrates the Commutative Property of Multiplication.

Step by step solution

01

Identify Law of Algebra

The equation given is \(6(7) = 7(6)\). Review the fundamental laws of algebra to identify the one that demonstrates the equality of these two expressions.
02

Recognize Commutative Property

Evaluate if swapping the positions of the numbers in the multiplication affects the outcome. In this case, multiplication of two numbers does not change the result if you switch them, which illustrates the **Commutative Property of Multiplication**.
03

Confirm Understanding

The commutative property states that for any two numbers \(a\) and \(b\), the equation \(a \cdot b = b \cdot a\) holds true. Here, \(6\) and \(7\) confirm this property, since \(6 \times 7 = 7 \times 6\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Laws of Algebra
The laws of algebra are fundamental rules that guide the manipulation and understanding of algebraic expressions. These laws include the commutative, associative, and distributive properties. They are like building blocks for solving algebra problems, providing a reliable framework to simplify and rearrange expressions.

Each law plays a unique role and comes into play in different mathematical operations:
  • Commutative Law: This applies to both addition and multiplication, allowing numbers to be swapped without affecting the result.
  • Associative Law: This law captures the grouping of numbers, indicating that when adding or multiplying, the way numbers are grouped does not change the outcome.
  • Distributive Law: This law combines addition and multiplication, allowing us to distribute a multiplication operation across an addition inside parentheses.
Understanding these laws helps in solving equations and rearranging terms to simplify complex algebra problems.
Multiplication
Multiplication is one of the basic arithmetic operations, like addition, subtraction, and division. It involves finding the total number when one number is taken repeatedly a certain number of times.

In the problem presented, multiplication is used to demonstrate the commutative property. The operation is simple: multiplying 6 by 7 means that you are adding the number 6, seven times, or vice versa. In either case:
  • The expression involves two factors: 6 and 7.
  • The product or the result will always be the same, regardless of the order of the numbers.
Multiplication forms the basis for more complex mathematical concepts and is crucial in various real-world applications.
Algebraic Properties
Algebraic properties play a crucial role in simplifying and solving equations. They are rules that govern the way numbers and variables interact in equations. Among these, the commutative property is one of the most intuitive and widely used across different scenarios.

For the commutative property, which is central to the given exercise:
  • It ensures the order in which two numbers are multiplied does not affect the overall product.
  • This property makes calculations more flexible and convenient because one can rearrange terms to find solutions more easily.
Understanding algebraic properties allows students to manipulate expressions confidently and solve complex problems efficiently. Mastery of these properties provides a strong foundation for further study in mathematics.

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