Problem 20 - Elementary Algebra

Question

Name the property of real numbers that makes \(4 y x^{2}=4 x^{2} y\) a true statement.

Step-by-Step Solution

Answer

Answer: The commutative property of multiplication.

Step 1: Identify the given equation

The given equation is \(4yx^2 = 4x^2y\).

Step 2: Apply the commutative property of multiplication

The commutative property of multiplication states that for any real numbers a and b, \(a*b = b*a\). In the given equation, we apply the commutative property to the two variables, \(y\) and \(x^{2}\). According to the commutative property of multiplication, we have \(yx^2 = x^2y\).

Step 3: Show that the given equation is true

Now, let's multiply both sides of the equation by the constant 4. \(4(yx^2) = 4(x^2y)\) Plugging in the simplified expressions from step 2, we get: \(4(4yx^2) = 4(4x^2y)\) This confirms that the given equation is indeed true.

Step 4: Conclusion

Therefore, the property of real numbers that makes the equation \(4yx^2 = 4x^2y\) true is the commutative property of multiplication.

Key Concepts

Real Numbers

Real numbers are a fundamental part of mathematics and encompass a vast range of numbers. They include:

• Whole numbers (0, 1, 2, 3, ...)
• Fractions and decimals
• Positive and negative numbers

Real numbers are represented by the symbol \( \mathbb{R} \). These numbers can be placed on a continuous number line, demonstrating their use in representing measurements or quantities in the real world.
They form the basis for various mathematical operations, including addition, subtraction, multiplication, and division. Understanding real numbers equips you with the ability to tackle a wide range of mathematical problems, especially those involving variable expressions and equations. They are not limited to counting numbers and also include complex constructs like square roots of positive numbers and irrational numbers such as \( \pi \) and \( e \).
In the context of the commutative property, real numbers serve as the elements which can be rearranged without affecting the overall result.

Multiplication

Multiplication is a basic arithmetic operation that involves combining groups of equal size. It's one of the four fundamental operations of mathematics, alongside addition, subtraction, and division.
In a multiplication operation, you have two numbers, known as factors, and the result of their multiplication is called the product. For instance, in the expression \(4 \times 3 = 12\), 4 and 3 are factors, and 12 is the product.

• The commutative property of multiplication tells us that the order of these factors doesn't matter when determining the product. Mathematically, this is represented as \( a \times b = b \times a \).
• This property holds true for all real numbers, essentially allowing flexibility in how we compute multiplications.

This principle is extremely useful in simplifying expressions, especially those involving multiple variables or numbers, which we encounter in variable expressions.

Variable Expressions

Variable expressions are combinations of numbers and variables using operations such as addition, subtraction, multiplication, and division. In such expressions, variables can represent unknown or changeable quantities.
Consider the expression \( 4yx^2 \). Here, \( y \) and \( x \) are variables, while 4 is a constant factor. The expression represents a general form that can change depending on the values assigned to \( y \) and \( x \).

• By applying properties like the commutative property, variable expressions can be rearranged without changing their value or meaning.
• The expression \( 4yx^2 = 4x^2y \) demonstrates the use of the commutative property in simplifying and verifying variable expressions.

Understanding variable expressions is crucial for solving equations and grasping more complex algebraic concepts. They allow you to write general cases and prepare for solving problems where one or more quantities are unknown.