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Chapter 5: Problem 65

Simplify it by combining any like terms. $$ 9.6 z-3.2-11.7 z-6.9 $$

Short Answer

Expert verified
The simplified expression is \(-2.1z - 10.1\).

Step by step solution

01

Identify like terms

To simplify the expression, first identify the terms that can be combined. The expression is \(9.6z - 3.2 - 11.7z - 6.9\). The like terms here are those involving \(z\): \(9.6z\) and \(-11.7z\). The constants \(-3.2\) and \(-6.9\) can also be combined.
02

Combine like terms involving z

Combine the \(z\) terms: \(9.6z - 11.7z = (9.6 - 11.7)z = -2.1z\). This simplifies the expression to \(-2.1z - 3.2 - 6.9\).
03

Combine constant terms

Combine the constant terms: \(-3.2 - 6.9 = -10.1\). Now, the expression is further simplified to \(-2.1z - 10.1\).

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Key Concepts

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

Understanding Like Terms
In algebra, the term "like terms" refers to terms that have the same variables raised to the same power. This means they can be combined through addition or subtraction. Think of them as similar items in a basket; only similar items can be grouped together.
For example, in the expression \(9.6z - 3.2 - 11.7z - 6.9\), the terms containing \(z\), which are \(9.6z\) and \(-11.7z\), are like terms. They all share the variable \(z\).
  • Like terms have identical variable parts.
  • They can have different coefficients (numbers in front of the variables).
  • They can always be added or subtracted, unlike terms that are completely different.
Recognizing like terms is a fundamental skill in algebra as it allows for simplifying expressions more easily.
Combining Like Terms
Once you identify like terms in an expression, the next step is combining them, which is an essential process in algebra simplification. Combining like terms means adding or subtracting their coefficients while keeping the variable part intact.
For instance, with our example, we have identified \(9.6z\) and \(-11.7z\) as like terms. To combine them, simply subtract their coefficients: \(9.6 - 11.7 = -2.1\). This means our like terms \(9.6z\) and \(-11.7z\) simplify to \(-2.1z\).
  • Combine terms with the same variable by adjusting their coefficients.
  • Use arithmetic operations (addition or subtraction) as indicated by their signs.
  • Simplifying the expression helps in solving algebraic equations more effectively.
Combining like terms results in a cleaner and easier-to-understand expression, which can be crucial when solving equations or analyzing expressions further.
Constants in Algebra
Constants in algebra are terms without any variables. They're simply fixed numerical values, like \(3.2\) and \(6.9\) in our expression. Despite lacking variables, constants can always be combined through basic arithmetic.
In our example, once we've simplified the like terms with \(z\), we are left with the constants \(-3.2\) and \(-6.9\). Adding or subtracting these constants follows the same rules as regular arithmetic: \(-3.2 - 6.9 = -10.1\).
  • Constants remain unchanged regardless of variable values.
  • They can be combined simply by performing addition or subtraction.
  • Simplification of constants often precedes other algebraic operations.
Understanding how to work with constants is a key skill that assists in making equations and expressions more manageable. After all, simplifying expressions is all about making them as straightforward as possible.

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