Problem 5 - Algebra for College Students

Question

Simplify the algebraic expressions in Problems \(1-14\) by combining similar terms. $$4 n-9 n-n$$

Step-by-Step Solution

Answer

The simplified expression is \(-6n\).

Step 1: Identify the Like Terms

In algebra, terms are considered 'like terms' if they have the same variable raised to the same power. In this expression, all terms \(4n, -9n,\) and \(-n\) are 'like terms' because they all have the variable \(n\).

Step 2: Combine Like Terms

To combine like terms, simply add their coefficients (the numerical part in front of the letter). For this expression: \(4n - 9n - n \) can be rewritten as:\((4 - 9 - 1)n\)Calculate the coefficient: \(4 - 9 - 1 = -6\)

Step 3: Write the Simplified Expression

Now, replace the coefficient of \(n\) in the expression:\(-6n\)This is the simplified form of the given expression.

Key Concepts

Like Terms

When working with algebraic expressions, identifying 'like terms' is crucial for simplification. Like terms are those terms in an expression that contain the same variables raised to the same power. For example, in the expression given, we have three terms: \(4n\), \(-9n\), and \(-n\). All these terms have the variable \(n\) without any exponents, hence they are like terms.

To put it simply, like terms make it easier to combine terms because they share common characteristics. In this way, the expression becomes more manageable, just like organizing ingredients by category when baking.

• Terms with the same variable are potential candidates for like terms.
• Check if they have the same exponent (if any) for all terms.

Combining Like Terms

Once you've identified the like terms in an expression, the next step is combining them. This means adding or subtracting the coefficients (the numbers in front of the variables) of those like terms. In the expression \(4n - 9n - n\), all terms are like terms since they all involve \(n\). Adding these terms involves:

• First, note the coefficients: 4, -9, and -1 (implicitly, since \(-n\) is \(-1n\)).
• Second, perform the arithmetic: \(4 - 9 - 1\).

By combining them, you calculate \(4 - 9 - 1 = -6\).

This process simplifies the expression to \(-6n\), which is clearer and more concise than the original expression. Remember to always combine only like terms to avoid errors.

Algebraic Coefficients

The algebraic coefficients are the numbers located in front of the variables in an algebraic expression. Their role is crucial as they determine how many times the associated term affects the equation. Looking at the expression \(4n - 9n - n\), the coefficients are 4, -9, and -1.

Understanding how to manipulate these coefficients is vital:

• To combine like terms, we need to focus on adding or subtracting these coefficients.
• Even if a term doesn't appear to have a coefficient, such as \(-n\), it is actually just hiding: here, it is truly \(-1n\).

When working with coefficients, always keep the sign in mind. This affects whether the term will increase or decrease in our final simplified expression. The coefficients provide the multiplier for the variable, shaping the term's influence in the expression.