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Chapter 14: Problem 94

Simplify. $$ \left(a_{1}+5 d\right)+\left(a_{n}-5 d\right) $$

Short Answer

Expert verified
The simplified expression is \(a_{1} + a_{n}\).

Step by step solution

01

Identify the expression components

The given expression is \((a_{1} + 5d) + (a_{n} - 5d)\). Identify each part of the expression.
02

Distribute the terms

Combine the like terms from both parts of the expression: \(a_{1} + 5d + a_{n} - 5d\).
03

Combine like terms

Notice that \(5d\) and \(-5d\) cancel each other out. Thus, the expression simplifies to \(a_{1} + a_{n}\).

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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 foundational skill in algebra. It involves rewriting an expression in its simplest form. For the given problem, we started with the expression \( (a_{1} + 5d) + (a_{n} - 5d) \). The goal is to reduce this to a simpler form. Simplification often involves combining like terms and using algebraic properties. By simplifying an expression, you make it easier to understand and solve.
Like Terms
Like terms are terms that have the same variables raised to the same power. They can be combined to simplify an expression. In the given problem, we see terms like \(5d\) and \(-5d\). These are like terms because both contain the variable \(d\). When we combine \(5d\) and \(-5d\), they cancel each other out, because \(5d - 5d=0\). What remains are the terms \(a_{1} \) and \(a_{n}\), which do not cancel each other out but are combined to give us the simplified expression \(a_{1} + a_{n}\).
  • Identifying like terms is crucial for simplification.
  • Like terms must have the same variables and exponents.
  • Once identified, like terms can be added or subtracted to simplify the expression.
Distributive Property
The distributive property is a useful algebraic property for simplifying expressions. According to the distributive property, \(a(b + c) = ab + ac\). In our problem, although we don't explicitly distribute terms, understanding this property helps in combining like terms as seen in the initial step of simplification, \(a_{1} + 5d + a_{n} - 5d\). Here, recognizing that \(5d\) and \(-5d\) are like terms allows us to cancel them out directly, simplifying our overall expression.
  • The distributive property allows for multiplication to be distributed over addition or subtraction within parentheses.
  • It helps simplify expressions and solve equations efficiently.
  • Knowing when and how to apply this property is key in algebra.

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