Problem 9 - Fundamentals of Mathematics

Question

Simplify each expression by combining like terms. $$11 k-8 k$$

Step-by-Step Solution

Answer

The simplified expression is \(3k\).

Step 1: Identify the Like Terms

In the expression given, identify the terms that are considered 'like terms'. Like terms are terms that have identical variable parts. In this case, both terms, \(11k\) and \(-8k\), contain the variable \(k\).

Step 2: Combine the Coefficients

Since the terms are like terms, we can combine them by adding their coefficients. The coefficients are the numerical multipliers of the variable \(k\). Here, the coefficients are 11 and -8.

Step 3: Perform the Addition

To combine the coefficients, perform the addition \(11 + (-8)\). This simplifies to \(3\), as \(11 - 8 = 3\).

Step 4: Write the Simplified Expression

Attach the combined coefficient to the variable \(k\). Therefore, the simplified expression is \(3k\).

Key Concepts

Identifying Like Terms

When simplifying algebraic expressions, the first step is identifying like terms. Like terms are terms within an expression that have exactly the same variable part, including the same powers. This is crucial because only like terms can be combined to simplify expressions.
For example, let's consider the terms in the expression given: \(11k - 8k\). Both \(11k\) and \(-8k\) are like terms because they both contain the variable \(k\), and there's no difference in the powers of \(k\) (they're both \(k^1\)). Here's how to spot like terms:

• Check that variables are identical.
• Verify that exponents of the variables are the same.

When you identify like terms in an expression, you're ready to combine them to simplify it.

Combining Coefficients

Once you have identified like terms, the next step is combining their coefficients. Coefficients are the numbers in front of the variables and dictate how many of each term you have.
In our example, \(11k - 8k\), the coefficients are 11 and -8. To combine the like terms, we simply add the coefficients together. The process for combining coefficients involves:

• Paying attention to the signs of the coefficients. This is essential because adding negative numbers is integral to simplification.
• Adding the coefficients together: \(11 + (-8)\).

By computing the sum of the coefficients, \(11 - 8\), we find it equals 3. So the combined coefficient in this case is 3.
Remember, combining coefficients correctly is key to maintaining the integrity of the algebraic expression.

Simplifying Expressions

The ultimate goal of identifying like terms and combining their coefficients is to simplify expressions. Simplifying means reducing an expression to its most concise form without changing its value.
For our worked example, once we combine the coefficients and derive a result of 3, we simply attach this back to the variable part. Since the variables were like, \(k\) stays consistent across terms, resulting in: \(3k\).Steps to simplify an expression:

• Identify all groups of like terms in the expression.
• Combine the coefficients of each group of like terms.
• Reattach the combined coefficient to the common variable.

The simplified expression, \(3k\), is much easier to use in further calculations or evaluations. Simplifying expressions not only makes them aesthetically cleaner but also makes manipulations in math more effortless.