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Chapter 5: Problem 42

Perform the indicated operations. See Examples 9 through 13 . $$ 4-(-y-4) $$

Short Answer

Expert verified
The simplified expression is \(8 + y\).

Step by step solution

01

Remove Parentheses

The expression given is: \(4 - (-y - 4)\). Begin by removing the parentheses. When you remove a parenthesis preceded by a minus sign, change the sign of each term inside the parentheses. Thus, it becomes \(4 + y + 4\).
02

Simplify the Expression

Combine like terms in the expression to simplify it. Combine the constants: \(4\) and \(4\). The expression simplifies to \(8 + y\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
Simplifying expressions is a fundamental skill in algebra that makes working with equations easier. It involves performing operations to condense an expression to its simplest form. This usually means reducing the number of terms and making the equation look neater, allowing you to see the relationships between its components more clearly.

To simplify an expression, follow these guidelines:
  • Combine constants and coefficients.
  • Eliminate any unnecessary parts, like extra numbers or variables.
  • Put the expression in the most straightforward order, often by grouping like terms together.
For example, when working with the expression from the problem, after rewriting it as \(4 + y + 4\), simplifying involves adding the numbers together to get a tidy \(8 + y\). This makes it easier to use in further calculations.
Like Terms
In algebra, identifying and combining like terms is crucial as it significantly reduces the complexity of expressions. Like terms are terms that have identical variable parts, meaning they contain the same variables raised to the same powers.

Here鈥檚 how to spot and combine like terms:
  • Check the variables and their exponents.
  • Combine their coefficients by adding or subtracting them.
  • For constant terms (numbers without variables), simply add or subtract them directly.
In the expression \(4 + y + 4\), "4" and "4" are like terms because they are both constants. So, you combine them to get \(8\) as part of simplifying the expression. This step significantly cuts down the complexity and potential for mistakes in later algebraic manipulations.
Removing Parentheses
Removing parentheses correctly in algebraic expressions is essential, especially when dealing with negative signs. When an expression includes parentheses, you often have to distribute or remove them to simplify.

Here's how to approach removing parentheses:
  • If there's a "-" sign before the parenthesis, distribute it by flipping the sign of each term inside.
  • If there's a "+" sign or no sign, keep the terms as they are.
  • Ensure each term is altered as needed based on the operation in front of the parenthesis.
For instance, in the problem \(4 - (-y - 4)\), removing the parentheses requires reversing the signs inside, turning it into \(4 + y + 4\). This operation is key to correctly simplifying expressions in many algebra problems.

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Most popular questions from this chapter

Multiply. $$(3 y-1)^{3}$$

In which of the following is \(\frac{a+7}{7}\) simplified correctly? See the Concept Check in this section. a. \(a+1\) b. \(a\) c. \(\frac{a}{7}+1\)

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