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Chapter 7: Problem 186

Simplify using the properties of identities, inverses, and zero. \(19 a+44-19 a\)

Short Answer

Expert verified
44

Step by step solution

01

- Identify like terms

In the expression, identify the like terms. Here, the like terms are the terms with the variable 'a': \(19a\) and \(-19a\).
02

- Apply additive inverse property

Combine the identified like terms using the additive inverse property: \(19a - 19a = 0\). This shows the terms cancel each other out.
03

- Include the constant term

After combining the like terms, include the remaining constant term in the expression. So, the simplified expression becomes: \(0 + 44\) or simply \(44\).

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Key Concepts

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

Properties of Identities
The properties of identities help us simplify algebraic expressions by providing a basis for equal and opposite values. The identity property of addition tells us that any number plus zero will yield that same number. For example, in the expression 19a + 44 - 19athe two terms with 'a' (19aand -19a) cancel each other due to the additive inverse property (more on that later). This leaves us with the identity property: 0 + 44equals 44. Therefore, understanding identity properties simplifies algebraic expressions significantly.
Additive Inverse Property
The additive inverse property is a key concept in simplifying algebraic expressions. This property states that for every number, there's another number that, when added together, equal zero. These pairs are called inverses. For instance, let's look at the expression again: 19a + 44 - 19a. The terms 19a and -19a can be seen as inverses because they add up to zero: 19a - 19a = 0. This principle helps us to simplify expressions effectively by eliminating terms that cancel each other out.
Zero Property in Algebra
The zero property is another fundamental concept in algebra. It's straightforward but immensely useful. Essentially, if you add zero to any number, the number remains unchanged. This property comes into play after we apply the additive inverse property. In our example, after reducing 19a + 44 - 19ato0 + 44, we use the zero property to further simplify it: 0 + 44 = 44. So, the zero property assures us that adding zero to any number doesn't change its value, allowing us to neatly conclude our simplification.

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