Problem 19 - Precalculus Mathematics for Calculus

Question

19-24 - 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: Identify the Expression

The given expression is \(3(x+y)\). This expression is a multiplication of \(3\) by the entire expression \((x+y)\).

Step 2: Use the Distributive Property

Apply the distributive property which states that \(a(b+c) = ab + ac\). In this case, \(a = 3\), \(b = x\), and \(c = y\).

Step 3: Distribute and Simplify

Using the distributive property: \[3(x+y) = 3 \cdot x + 3 \cdot y = 3x + 3y\].

Step 4: Present the Final Expression

The expression without parentheses is \(3x + 3y\). This is the simplified form without any grouping symbols.

Key Concepts

Real Numbers

Real numbers are the set of numbers that include every type of number you have encountered in mathematics, including integers, fractions, and decimals. They can be both positive and negative and come in many forms:

• Integers: Whole numbers that can be positive or negative, like -3, 0, and 7.
• Rational Numbers: Numbers that can be expressed as a fraction of two integers, such as \(\frac{1}{2}\) or 4.5.
• Irrational Numbers: Numbers that cannot be expressed as a simple fraction, such as \(\sqrt{2}\) or \(\pi\).

This exercise involves manipulating real numbers when applying the distributive property. Understanding the nature of real numbers is essential as they play a vital role when simplifying expressions or solving algebraic equations. To simplify expressions like the one given \(3(x+y)\), we first apply our knowledge of how real numbers interact with each other, particularly how they are added and multiplied.

Multiplication

Multiplication is one of the fundamental operations in mathematics. It is the process of adding a number to itself repeatedly. For example, \(3 \times 4\) means adding 3 a total of 4 times, which equals 12.When multiplying numbers or expressions, we often use the distributive property, which serves to simplify expressions and eliminate parentheses. In this context, multiplication involves distributing a single term multiplier over a sum or difference within parentheses.In our expression \(3(x+y)\), the number 3 is multiplied separately by each term inside the parentheses:

• 3 and x multiply to give \(3x\).
• 3 and y multiply to give \(3y\).

Thus, these multiplication steps combined provide us the simplified form \(3x + 3y\). Multiplication of real numbers follows the commutative, associative, and distributive properties, which make calculations systematic and consistent across algebraic expressions.

Simplifying Expressions

Simplifying expressions generally involves rewriting them in a simpler form by eliminating any unnecessary parts like parentheses or combining like terms. It involves using properties of mathematics like the distributive property, which helps break larger expressions into smaller, more manageable parts.To simplify the given expression \(3(x+y)\), we used the distributive property to remove the parentheses. The steps include:

• Recognizing the multiplication across the bracket as \(3 \cdot (x+y)\).
• Applying the distributive property: \(3 \cdot x + 3 \cdot y\).
• Finally, writing it as \(3x + 3y\), the simplified version without any parentheses.

Simplifying expressions is crucial because it makes them easier to work with, especially when solving equations or comparing mathematical quantities. This process reduces complexity and helps in clearly understanding the mathematical relationships within the expressions.