Problem 18 - Precalculus Mathematics for Calculus

Question

Rewrite the expression using the given property of real numbers. Distributive Property, \(5 x+5 y=\) _____

Step-by-Step Solution

Answer

The expression using the Distributive Property is \(5(x+y)\).

Step 1: Recognize the Distributive Property

The Distributive Property of real numbers states that for any numbers \(a, b,\) and \(c\), the expression \(a(b + c)\) is equal to \(ab + ac\). In our problem, we are given the expression \(5x + 5y\) and need to rewrite it using this property.

Step 2: Identify the Common Factor

Look for a common factor in the terms \(5x\) and \(5y\). In this case, the common factor is \(5\) since both terms are divisible by 5.

Step 3: Apply the Distributive Property

Factor out the common factor \(5\) from each term and express the original equation as a product: \(5(x + y)\). This is done by rewriting \(5x + 5y\) as \(5(x + y)\), which shows the use of the Distributive Property.

Key Concepts

Understanding Real Numbers

Real numbers are a fundamental part of mathematics, encompassing all the numbers you might encounter in everyday life. This category includes:

• Integers: Both positive and negative whole numbers, including zero (e.g., -3, 0, 4).
• Rational numbers: Numbers that can be written as a fraction, such as \( \frac{2}{3} \) or 0.75.
• Irrational numbers: Numbers that cannot be written as a simple fraction, like \( \sqrt{2} \) or π.
• Whole numbers: Non-negative integers, like 0, 1, 2, etc.
• Natural numbers: Positive integers, starting from 1, like 1, 2, 3, etc.

Real numbers are important because they form the building blocks of various mathematical operations. They allow us to use properties like the Distributive Property, enabling us to simplify and manipulate equations. Recognizing parts like common factors within real numbers helps apply these concepts effectively when working through algebraic expressions.

Factoring in Algebra

Factoring is a method used in algebra to simplify expressions or solve equations by breaking them down into products of simpler expressions. When you factor an expression, you are essentially looking for numbers or expressions that multiply together to give the original expression. This process involves identifying common factors in terms. For instance, consider the expression \(5x + 5y\).The process of factoring this equation consists of:

• Recognizing the common factor that appears in both terms, which in this case is 5.
• Rewriting the expression by dividing each term by the common factor and placing the factor outside the brackets. Thus, the equation \(5x + 5y\) becomes \(5(x + y)\).

Factoring is an essential skill that not only simplifies equations but also plays a major role in solving quadratic equations, simplifying fractions, and enabling the use of the Distributive Property in reverse to expand expressions.

Identifying the Common Factor

A common factor is a number or algebraic expression that divides two or more terms evenly. Finding a common factor is a key step in factoring expressions, especially in utilizing the Distributive Property. Let's look into why this is important using the expression \(5x + 5y\).In this expression:

• Observe that both terms, \(5x\) and \(5y\), are divisible by 5.
• The number 5 is the common factor because it is present in both terms, allowing you to factor it out.

By dividing each term by this common factor and rearranging the expression, you effectively simplify it from \(5x + 5y\) to the factored form \(5(x + y)\). This demonstrates the use of the Distributive Property, where the common factor is crucial in streamlining algebraic expressions for easier manipulation and understanding.