Problem 52 - Fundamentals of Mathematics

Question

For problems \(47-56\), simplify each expression by combining like terms. $$ 6 n-2 n+6-2-n $$

Step-by-Step Solution

Answer

The simplified expression is \(3n + 4\).

Step 1: Identify Like Terms

First, we identify the like terms in the expression. Like terms have the same variable raised to the same power. In the expression \(6n - 2n + 6 - 2 - n\), the like terms involving \(n\) are \(6n\), \(-2n\), and \(-n\). The constant terms are \(6\) and \(-2\).

Step 2: Combine Like Terms with n

Now, combine the like terms that involve \(n\). Add them together: \(6n - 2n - n\). This simplifies to \((6 - 2 - 1)n = 3n\).

Step 3: Combine Constant Terms

Next, combine the constant terms. The constants are \(6\) and \(-2\). Add them together: \(6 - 2 = 4\).

Step 4: Formulate Final Expression

Now, replace the initial expression with the simplified terms. We have \(3n\) from the like terms of \(n\) and \(+4\) from the constants. The simplified expression is \(3n + 4\).

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, one crucial step is combining like terms. Like terms are those that have identical variable parts, such as the same letters raised to the same power. For example, in the expression \(6n - 2n + 6 - 2 - n\), the terms \(6n\), \(-2n\), and \(-n\) are like terms because they all contain the variable \(n\) raised to the first power.
To combine like terms, simply add or subtract the coefficients (the numbers in front of the variables). In our example, this means we calculate \(6n - 2n - n\). First, combine \(6n\) and \(-2n\), which equals \(4n\). Then, subtract \(n\) from \(4n\) to get \(3n\).
Combining like terms makes the expression simpler and easier to work with. Always remember:

• Identify terms with the same variables and powers.
• Add or subtract their coefficients.
• Write down the simplified term with its coefficient and variable.

Algebraic Expressions

An algebraic expression is a mathematical statement that includes numbers, variables, and operations (like addition or subtraction). Expressions can become quite complex, but the goal in many math problems is to simplify them as much as possible.
Take for example the expression \(6n - 2n + 6 - 2 - n\). It is composed of:

• Variables (like \(n\))
• Coefficients (like \(6\), \(-2\), and \(-1\) in front of \(n\))
• Constant terms (like \(6\) and \(-2\))

Algebraic expressions don't usually have equality signs (\(=\)), unless they are part of an equation. In this exercise, the expression is a combination of terms, and our task is to simplify it by combining like terms and constants. This helps us express the information or relationship more clearly.
When dealing with algebraic expressions, remember:

• Identify different parts of the expression: variables, coefficients, and constants.
• Perform operations such as addition or subtraction to simplify.
• Write simplified forms to make calculations easier.

Constant Terms

Constant terms are numbers in an expression that stand alone, without attached variables. They represent fixed values and are important parts of simplifying expressions. In the expression \(6n - 2n + 6 - 2 - n\), the constant terms are \(6\) and \(-2\).
To simplify these constants, you perform basic arithmetic operations. Here, add \(6\) and \(-2\) to get \(4\). This step is as vital as combining like terms with variables because it simplifies the entire expression.
Key points to remember about constant terms are:

• They do not change when the variable does.
• They can be added or subtracted directly.
• Combining them simplifies the expression further.

Constant terms make expressions easier to handle, as they reduce the number of components you need to consider in calculations.