Problem 31 - Algebra A Combined Function

Question

Use the distributive property to write each expression without parentheses. Then simplify the result, if possible. See Examples 7 through 12. $$ 4(x+y) $$

Step-by-Step Solution

Answer

The expression \(4(x+y)\) simplifies to \(4x + 4y\).

Step 1: Apply the Distributive Property

The distributive property states that for any numbers or expressions \(a\), \(b\), and \(c\), \(a(b + c) = ab + ac\). In this problem, we have \(4(x+y)\). Apply the distributive property to remove the parentheses by multiplying \(4\) with \(x\) and \(4\) with \(y\). This gives us \(4x + 4y\).

Step 2: Simplify the Expression

After applying the distributive property, the expression obtained is \(4x + 4y\). There are no like terms to combine in this expression, so it is already in its simplest form.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are a fundamental concept in algebra, consisting of numbers, variables, and operators (such as addition and multiplication). They can represent real-world problems succinctly and allow us to perform various operations to find unknown values or simplify them. The expression given in the original exercise, \(4(x+y)\), is an example of an algebraic expression. Here:

• The number \(4\) is called the coefficient.
• \(x\) and \(y\) are variables representing unknown values.
• The operations included are addition and multiplication.

Expressions can appear complex, but they follow rules and properties—like the distributive property—that make it easier to work with them. Recognizing that \(4(x+y)\) involves distribution hints to the process of rewriting it more simply, aiding in both calculation and deeper algebraic understanding.

Simplification in Algebra

Simplification is the process of transforming a mathematical expression into a simpler or more readable form without changing its value. It helps to make calculations straightforward and the expressions easier to understand. In algebra, simplification involves several key steps:

• Applying algebraic properties, such as distributive, associative, and commutative properties.
• Combining like terms (if any).
• Rearranging terms to achieve the simplest form.

In the solution provided for \(4(x+y)\), the distributive property was used to simplify the expression to \(4x + 4y\). Simplification here was straightforward because there are no like terms to combine. Always remember, simplification not only makes problem-solving efficient but also helps reduce errors in calculations.

Identifying and Combining Like Terms

Like terms in algebraic expressions are terms that have identical variable parts, making them eligible to be combined. Identifying these is essential for simplifying expressions further. Here’s how you know they are like terms:

• They have the exact same variables raised to the same powers.
• Only their coefficients differ.

In the expression \(4x + 4y\) from our example, \(4x\) and \(4y\) are not like terms because though they each have coefficients of 4, they involve different variables. Hence, they cannot be combined further to simplify the expression. Mastering the concept of like terms is crucial for algebraic manipulations, and correctly combining them leads to a cleaner, built-for-solution expression.