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Chapter 2: Problem 6

Simplify each expression by combining any like terms. See Examples 3 and 4. $$ 6 g+5-3 g-7 $$

Short Answer

Expert verified
The simplified expression is \(3g - 2\).

Step by step solution

01

Identify like terms

Look at the expression and identify the like terms. In this case, the terms with the variable 'g' are the like terms: \(6g\) and \(-3g\). The constants, \(5\) and \(-7\), are also like terms.
02

Combine like terms with 'g'

Combine the coefficients of like terms with the variable 'g'. The coefficients are \(6\) and \(-3\), so we add them together: \(6 - 3 = 3\). This gives us \(3g\).
03

Combine constant terms

Now, combine the constants \(5\) and \(-7\). Add these together: \(5 - 7 = -2\).
04

Form the simplified expression

Put together the simplified terms from Steps 2 and 3. The simplified expression is \(3g - 2\).

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Key Concepts

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

Like Terms
In algebra, like terms are terms that have the same variable raised to the same power. Even though they may have different coefficients, the key is the variable part, which must match. For example, in the expression \(6g + 5 - 3g - 7\), the terms \(6g\) and \(-3g\) are considered like terms because they both contain the variable \(g\). Similarly, the numbers \(5\) and \(-7\) are like terms because they are both constant terms without any variables.

Recognizing like terms is crucial because it allows you to simplify expressions. When an expression is simplified, it can be easier to work with, especially when solving equations or evaluating functions.
Simplifying Expressions
Simplifying expressions is all about making the expression easier to understand and work with. This process often involves combining like terms, reducing the expression to the smallest possible number of terms. In the given expression \(6g + 5 - 3g - 7\), we start by grouping and simplifying like terms.

For simplification:
  • Identify all the like terms.
  • Combine these like terms by performing the operations, such as addition or subtraction, indicated in the expression.
  • Ensure the expression is in its simplest form, meaning there are no further operations needed between like terms.
The result is a cleaner, more concise expression that maintains the same value as the original.
Combining Coefficients
To combine coefficients effectively, first focus on the parts of the expression that have the same variables. Coefficients are the numerical parts of terms, so in \(6g + 5 - 3g - 7\), the coefficients are \(6\) from \(6g\) and \(-3\) from \(-3g\). Combining these involves simply adding or subtracting the coefficients: \(6 - 3 = 3\). Thus, the like terms with \(g\) become \(3g\).

Always remember that the operation sign (plus or minus) in front of the coefficient plays a crucial role when you combine them. Similarly, for constant terms like \(5\) and \(-7\), just perform normal addition or subtraction: \(5 - 7 = -2\). This approach ensures accurate simplification of expressions.

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