Problem 8 - Prealgebra

Question

Simplify expression. \(6 c+4+c+8\)

Step-by-Step Solution

Answer

The simplified expression is \(7c + 12\).

Step 1: Identify Like Terms

In the expression \(6c + 4 + c + 8\), first recognize the terms that are similar. We have two types of terms: those with the variable \(c\) and constant terms without variables. The like terms here are \(6c\) and \(c\), as well as the constants \(+4\) and \(+8\).

Step 2: Combine Like Variables

Add the coefficients of the like terms involving \(c\). Here we combine \(6c\) and \(c\), which are both terms with the variable \(c\): \[ 6c + c = 7c \]

Step 3: Combine Constant Terms

Now, add the constant numbers: \(4\) and \(8\). This will help simplify the constants in the expression:\[ 4 + 8 = 12 \]

Step 4: Write the Simplified Expression

After combining the like terms, rewrite the expression by placing the terms together: \[ 7c + 12 \]This is the simplified version of the original expression.

Key Concepts

Like Terms

When working with algebraic expressions, one important concept is understanding the idea of like terms. Like terms are parts of an expression that have the same variable raised to the same power. They are crucial because only like terms can be combined to simplify an expression. For example, in the expression \(6c + 4 + c + 8\), the like terms include the terms with the variable \(c\), which are \(6c\) and \(c\). Additionally, the constant numbers \(+4\) and \(+8\) are also like terms because they don't include variables.

Recognizing like terms is the first step in the journey to simplify algebraic expressions, as emphasizing commonalities among terms allows students to streamline and condense expressions efficiently. This focused approach transforms seemingly complex algebra into more manageable pieces by allowing multiple terms to be combined into a single, simpler term.

Combining Coefficients

Once you identify like terms, it's time to combine the coefficients of those terms. Coefficients are the numerical part of terms, such as the \(6\) in \(6c\), or the \(4\) and \(8\) in constants. When we say 'combine coefficients,' we're really talking about performing operations with the numbers in front of the variables similar to basic arithmetic.

In the given expression \(6c + 4 + c + 8\), you look at the coefficients of the terms with the variable \(c\). Adding the coefficients \(6\) and the invisible \(1\) in \(c\) gives us \(6c + c = (6 + 1)c = 7c\). Similarly, for the constant terms, you simply add the \(4\) and \(8\), which results in \(12\).

• This step is about purely numeric calculations and follows standard arithmetic rules.
• Understanding this process helps you to better handle and simplify more complex algebraic expressions.

This way, you gain not only the ability to simplify, but also confidence in manipulating algebraic terms.

Algebraic Expressions

Algebraic expressions are a combination of variables, numbers, and operations (like addition and subtraction). They're the foundation of algebra and can represent real-world situations or abstract mathematical concepts.

An example of an algebraic expression is \(6c + 4 + c + 8\). The purpose of simplifying an expression such as this one is to reduce it to its most streamlined form without changing its value, which makes it easier to analyze or solve.

In simplifying expressions, you break down complex or lengthy expressions by applying rules like those of like terms and coefficient combination and transform them into simpler forms, such as \(7c + 12\) in this example. This not only prepares you for solving equations but also helps in understanding more advanced mathematics. Understanding algebraic expressions and how to manipulate them equips you with essential tools for both everyday situations and higher-level math.