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Chapter 1: Problem 83

Simplify each expression. See Examples \(11,14,\) and \(15 .\) $$ -(n+5)+(5 n-3) $$

Short Answer

Expert verified
The expression simplifies to \(4n - 8\).

Step by step solution

01

Distribute the Negative Sign

Start by distributing the negative sign across the terms inside the parentheses. The expression \(- (n + 5)\) becomes \(-n - 5\). Thus, the entire expression becomes \(-n - 5 + (5n - 3)\).
02

Distribute through the Second Parentheses

Since the second set of parentheses is preceded by a positive sign, you can remove them without changing the signs of the terms. The expression remains \(5n - 3\). The expression now is: \(-n - 5 + 5n - 3\).
03

Combine Like Terms

Combine the like terms in the expression. The terms with \( n \) are \(-n + 5n\), which equals \(4n\). The constant terms are \(-5 - 3\), which sum to \(-8\). Thus, the expression simplifies to \(4n - 8\).

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Key Concepts

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

Distributive Property
The distributive property is a key element in simplifying algebraic expressions, especially when dealing with parentheses. It states that every term inside the parentheses is multiplied by the term outside of it. In this exercise, the expression begins as \(-(n + 5) + (5n - 3)\). The negative sign in front of the first set of parentheses needs to be distributed across each term inside:
  • The expression \(-(n + 5)\) transforms to \(-n - 5\).
  • This means both the \(n\) and the \(+5\) are each multiplied by \(-1\), leading to \(-n\) and \(-5\), respectively.
Using the distributive property properly ensures that you change the signs correctly and make accurate calculations. Be careful with negative signs; they can change the outcome of the expression if not correctly distributed.
Like Terms
Like terms are terms in an algebraic expression that have the same variable raised to the same power. Recognizing and combining like terms is essential to simplifying expressions. In the simplified expression, we look to
  • Group the terms with the variable \(n\), namely \(-n\) and \(+5n\).
  • Combine these to get a single term, which is \(4n\).
In addition to terms including variables, like terms can also be constant terms:
  • The constants in the expression \(-n - 5 + 5n - 3\) are \(-5\) and \(-3\).
  • Combining these gives \(-8\).
Understanding how to correctly identify and group like terms is crucial for achieving simplified expressions efficiently and accurately.
Parentheses in Algebra
Parentheses in algebra often indicate terms that should be treated as a single unit or terms that require specific attention due to operations like multiplication or distribution. In the original expression, the role of parentheses is crucial:
  • The first part of the expression, \(- (n + 5)\), uses parentheses to show that both \(n\) and \(+5\) are affected by \(-1\).
  • The absence of a sign preceding specific operations implies that they should be executed last or outside the order of operations, ensuring accurate simplification.
Removing parentheses can often simplify an expression. However, care must be taken to maintain accurate operations, especially with negative signs. Knowing when and how to apply operations inside parentheses is a fundamental skill in algebra, aiding in clarity and accuracy when simplifying expressions.

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Most popular questions from this chapter

Simplify each expression. See Examples \(11,14,\) and \(15 .\) $$ -4(y z+3)-7 y z+1+y^{2} $$

Choose the fraction (s) equivalent to the given fraction. (There may be more than one correct choice.) $$ \begin{aligned} &\frac{5}{-(x+y)} \quad \text { a. } \frac{5}{(x+y)} \quad \text { b. } \frac{-5}{(x+y)}\\\ &\frac{-1}{-(x+y)} \cdot \frac{-5}{(x+y)} \end{aligned} $$

Write each of the following as an algebraic expression. See Examples 12 and 13. Two numbers have a sum of \(25 .\) If one number is \(x,\) represent the other number as an expression in \(x\).

Simplify each expression. See Examples \(11,14,\) and \(15 .\) $$ \frac{1}{3}(6 x-33 y)-\frac{1}{8}(24 x-40 y+1)-\frac{1}{3} $$

Translating Write each phrase as an algebraic expression. Use the variable \(x\) to represent each unknown number. See Example 8 . Fifteen and seven-tenths plus a number
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