Problem 102 - College Algebra

Question

Write each algebraic expression without parentheses. \(\frac{1}{2}(2 y)+[(-7 x)+7 x]\)

Step-by-Step Solution

Answer

The algebraic expression without parentheses is \(y\)

Step 1: Simplify the Fraction and Variable Multiplication

First, resolve the operation within the first set of parenthesis, which is multiplication: \(\frac{1}{2} * 2y\). Simplifying this will become \(y\).

Step 2: Resolve the Addition within the Square Brackets

Next, handle the operation within the square brackets [(-7x) + 7x]. Since -7x + 7x equals 0, these terms cancel each other out.

Step 3: Combine the Results from Step 1 and Step 2

Combine the result from step 1 and step 2. Since the term in the brackets equals 0, the final expression is \(y + 0\), which simplifies to \(y\)

Key Concepts

Parenthetical Operations in Algebra

When simplifying algebraic expressions, understanding how to deal with parenthetical operations, often seen as parentheses or brackets, is crucial. Parentheses and brackets indicate that the operations within them should be performed first, according to the order of operations, sometimes remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).

In the exercise \(\frac{1}{2}(2 y)+[(-7 x)+7 x]\), we first focus on the operation within the parentheses: \(\frac{1}{2}(2 y)\). Here, the multiplication inside the parentheses is resolved. After this step, attention is shifted to the square brackets: the terms inside can be either like terms - which can be combined - or different terms. It is important to simplify any parenthetical expression before moving on to the overall equation.

Combining Like Terms

When you come across terms in an algebraic expression that share the same variable and exponent, they are 'like terms' and can be combined. This simplification involves adding or subtracting the numerical coefficients while keeping the variable part unchanged.

For the exercise, within the square brackets \[(-7 x)+7 x\], we have two like terms with opposing coefficients, -7 and +7. These terms are combined by adding the coefficients, which in this case results in zero. Remember, combining like terms is a fundamental step to simplify expressions and make them more manageable.

Algebraic Expression Simplification

The goal of algebraic expression simplification is to make the expression as concise as possible while retaining its original value. This often involves executing parenthetical operations, combining like terms, and applying basic algebraic operations such as addition and subtraction.

Following the simplified parts from the initial expression, we've independently simplified \(\frac{1}{2}(2 y)\) to \(y\) and found that \[(-7 x)+7 x\] reduces to 0. The final step is to combine these results, which simply involves adding \(y\) and 0 together. The process not only simplifies the given algebraic expressions but also teaches valuable problem-solving strategies for future algebraic challenges.

Basic Algebraic Operations

The cornerstone of algebra is its basic operations: addition, subtraction, multiplication, and division. These must be performed following the correct order of operations. Multiplication and division are on the same hierarchical level and should be carried out from left to right, followed by addition and subtraction.

For example, our initial operation \(\frac{1}{2}(2 y)\) consists of a multiplication, which is executed before any additions or subtractions. After combining like terms, no further operations are present, and the end result is a single term expression. Mastery of these basic operations enables the simplification of more complex equations and is the foundation upon which more advanced algebraic concepts are built.