Problem 13 - Algebra A Combined Function

Question

Simplify each expression by combining any like terms. $$ 2 k-k-6 $$

Step-by-Step Solution

Answer

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

Step 1: Identify Like Terms

In the expression \(2k - k - 6\), the like terms are \(2k\) and \(-k\). Both terms contain the variable \(k\). The number \(-6\) is a constant and has no like term to combine with.

Step 2: Combine the Like Terms

Combine the coefficients of \(k\). Start with \(2k\). If you subtract \(k\), you remainder with \(2k - 1k = 1k\) which simplifies to \(k\). The expression becomes \(k - 6\).

Step 3: Final Simplified Expression

The simplified expression, after combining like terms, is \(k - 6\). This is as simplified as the expression gets since there are no additional like terms.

Key Concepts

Understanding Like Terms

In algebra, identifying "like terms" is crucial for simplifying expressions. Like terms are terms that have the same variable raised to the same power. Essentially, you can combine them because they are similar.

• The concept revolves around the variable part: the variable must be the same.
• For example, in the expression \(2k - k - 6\), the terms \(2k\) and \(-k\) are like terms because they both involve the variable \(k\).
• Constants, like \(-6\) in our expression, are not like terms with variable terms. They can only combine with other constants.

To simplify expressions involving like terms, combine them by adding or subtracting their coefficients. This process helps to reduce the expression to its most compact form, making it easier to work with in further algebraic operations.

What are Coefficients?

The term "coefficient" refers to the number that is in front of a variable in an algebraic term. It's the numerical factor that multiplies the variable. To easily spot a coefficient, simply look for the number directly sitting in front of a variable.

• In the expression \(2k\), the coefficient is \(2\).
• For the term \(-k\), the coefficient is \(-1\). This is because \(-k\) is implicitly \(-1 \times k\).

When combining like terms, you essentially add or subtract their coefficients while keeping the variable part constant. This results in a simplified expression. Recognizing and manipulating coefficients is a key skill for combining like terms effectively.

Breaking Down Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operations. They can range from simple, like \(2x + 3\), to more complex forms. Understanding how to break down and simplify them is a fundamental step in algebra.

• Variables in an expression represent unknown quantities, often denoted as letters (e.g., \(x\), \(k\)).
• Operations in expressions include addition, subtraction, multiplication, and division.
• When faced with simplifying an algebraic expression, first look for like terms to combine.
• Next, pay attention to the coefficients, as they dictate how much of each variable is present to combine or simplify.

By practicing this process, you become adept at efficiently handling expressions in algebra, modeling real-world situations, and solving mathematical problems.