Problem 23 - Algebra and Trigonometry

Question

Use properties of real numbers to write the expression without parentheses. 3\((x+y)\)

Step-by-Step Solution

Answer

The expression without parentheses is \(3x + 3y\).

Step 1: Recognize the property to use

To eliminate the parentheses in the expression \(3(x+y)\), we apply the Distributive Property. This property states that \(a(b+c) = ab + ac\).

Step 2: Apply the Distributive Property

Use the Distributive Property on \(3(x+y)\). Multiply \(3\) by each term inside the parentheses: \(3 imes x + 3 imes y\).

Step 3: Simplify the expression

Simplify the expression from Step 2: \(3x + 3y\). This is the expression without parentheses.

Key Concepts

Properties of Real Numbers

Understanding the properties of real numbers is essential for manipulating algebraic expressions. One critical property, particularly when dealing with expressions involving addition and multiplication, is the Distributive Property. The Distributive Property allows you to multiply a single term by each term inside a set of parentheses. For example, when you have an expression like \(a(b + c)\), this property tells us that it can be rewritten as \(ab + ac\).

By grasping this property, you can simplify expressions and solve equations more efficiently. This is because it converts a multiplication over addition into terms that can be more easily combined or rearranged. It combines elements of both the Associative and Commutative Properties to give this flexible simplification tool an established place in algebra, helping you to understand and tackle more complex mathematical problems.

Simplifying Expressions

Simplifying expressions is a key skill in algebra, helping you to make complex problems more manageable. The goal of simplification is to rewrite an expression in its most concise form while maintaining its original value. In the context of the Distributive Property, you simplify an expression by expanding it, which means removing the parentheses and combining like terms if necessary.

Taking the expression \(3(x + y)\) as an example, by applying the Distributive Property, it becomes \(3x + 3y\). This step makes the expression simpler and easier to handle. Remember, during simplification, small arithmetic operations such as adding or multiplying numbers are performed. These steps are crucial to ensure that you are left with an expression that's not only clean but also ready for further operations, aiding in solving equations or interpreting mathematical relationships.

Parentheses Elimination

Parentheses often group terms in algebra to indicate operations that should be prioritized. However, solving and simplifying expressions often require "eliminating" these parentheses at some point. This is where properties like the Distributive Property come into play. They allow us to methodically remove parentheses by distributing a factor outside across terms inside.

Let's look at how this works with the expression \(3(x + y)\). The operation inside the parentheses is \((x + y)\), and by applying the Distributive Property, we distribute the \(3\) to both \(x\) and \(y\), resulting in \(3x + 3y\).

The result is an expression free of parentheses, making it more straightforward and easier to solve in subsequent mathematical operations. Practice with parentheses elimination will boost your confidence and enhance your algebraic problem-solving skills.