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Chapter 7: Problem 187

Simplify using the properties of identities, inverses, and zero. \(27 c+16-27 c\)

Short Answer

Expert verified
16

Step by step solution

01

Identify and Group Like Terms

First, look at the given expression: \[27c + 16 - 27c\]. Identify like terms. Here, the terms \(27c\) and \(-27c\) are like terms because they contain the same variable \(c\).
02

Apply the Addition Inverse Property

Add the like terms \(27c\) and \(-27c\) together. The addition inverse property states that \(a + (-a) = 0\). Thus, \[27c - 27c = 0\].
03

Simplify the Expression

Substitute \(0\) for \(27c - 27c\) in the original expression and simplify. So, \[0 + 16\] simplifies to \[16\].

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Key Concepts

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

Addition Inverse Property
The addition inverse property is a very useful tool in algebra. The basic idea is simple: for any number or term, if you add its opposite, you get zero. Let's say you have a number like 5. Its opposite (or additive inverse) is -5. When you add 5 and -5 together, you get:

\[5 + (-5) = 0\]

This principle is also true for algebraic terms. In the problem we are solving, we apply this property to the terms \(27c\) and \(-27c\). Since they are opposites, adding them cancels them out and leaves zero:

\[27c + (-27c) = 0\]

This property helps us to simplify expressions by removing pairs of opposite terms.
Identifying Like Terms
Identifying like terms is crucial when simplifying algebraic expressions. Like terms have exactly the same variable parts. For example, in the expression \(27c + 16 - 27c\):
  • \(27c\) and \(-27c\) are like terms because they both contain the variable \(c\)
  • 16 does not have any variable, so it stands alone

To simplify, you group the like terms together. In this case:

\[27c - 27c + 16\]

After grouping the like terms, it's easier to apply the addition inverse property to simplify the expression further.
Simplification
Simplification of an algebraic expression means making it as simple as possible. After identifying and grouping like terms, and applying properties like the addition inverse property, we get a simpler form.
Let's follow through with the original problem again:
  • The given expression is: \(27c + 16 - 27c\).
  • Identified like terms: \(27c\) and \(-27c\).
  • Using addition inverse property: \(27c - 27c = 0\).
  • Substitute back into the expression: \(0 + 16 = 16\).

The result is 16, which is the simplified form. Simplification helps in solving problems efficiently by reducing complexity.

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