Problem 12 - Algebra 1: Concepts and Skills

Question

Find the terms of the expression. $$ 5 w-8 $$

Step-by-Step Solution

Answer

The terms of the expression \(5w - 8\) are \(5w\) and \(-8\).

Step 1: Identify the terms

The expression \(5w - 8\) can be rewritten as \((5w) + (-8)\).

Step 2: List the terms

The terms are \(\boxed{5w}\) and \(\boxed{-8}\).

Key Concepts

Terms of an Expression

In algebra, understanding the structure of an expression is vital. An expression is made up of parts called \textbf{terms}. Terms are the elements within an algebraic expression separated by plus (+) or minus (\textendash) signs. For example, in the expression \( 5w - 8 \) there are two terms: \(5w\) and \(\textendash8\).

• \textbf{\(5w\)}: This term consists of a coefficient and a variable. Here, the coefficient is \textbf{5}, and the variable is \textbf{w}. Coefficients are numbers that multiply the variable. In this context, \textbf{w} stands for a certain quantity that can vary, hence it's called a variable.
• \textbf{\(\textendash8\)}: This term is a constant because it does not contain a variable. It's simply a number on its own.

Each term plays a unique role in the algebraic expression, influencing how it behaves during calculations and when the value of the variable changes.

Variables in Algebra

Variables are the alphabet soup of algebra; they're symbols that represent unknown numerical values and are usually denoted by letters such as x, y, z, or, in our exercise, \(w\). Variables allow us to write expressions and equations that can apply to a variety of situations, not just one specific case.

• \textbf{Coefficients}: Variables can be accompanied by coefficients. In \( 5w \) from our original problem, the number \textbf{5} is the coefficient, showing that \(w\) is multiplied by \textbf{5}.
• \textbf{Usage in Expressions}: Variables let us work with expressions like \(5w - 8\) generally without needing to know the value of \(w\) right away. Only when we want to solve the expression for a particular value do we substitute it into the place of the variable.

Variables are the cornerstone of algebra, providing the flexibility to calculate and solve problems in an abstract yet structured way.

Subtracting Integers

Subtracting integers can sometimes seem tricky, but when you understand the rules, it's just as straightforward as any basic arithmetic. In subtracting integers:

• \textbf{Keep Change Change Rule}: This catchy phrase helps remember the process: 'Keep the first number the same, change the subtraction sign to an addition sign, and change the second number to its opposite.' For instance, if you have \(5 - (-8)\), you would change it to \(5 + 8\).
• \textbf{Understanding Negative Results}: When subtracting a larger number from a smaller one, the result will be negative. This is because you're essentially moving left on the number line rather than right, as you are subtracting value. In the expression \( 5w - 8 \), if \(w\) were to be 0, we'd simply have \( 0 - 8\), yielding \(\textendash8\).

The concept of subtracting integers is fundamental when we combine like or unlike terms in algebraic expressions, and it supports the framework for more complex algebraic manipulations.