Problem 18 - Fundamentals of Mathematics

Question

Specify each term. $$8 s+2 r-7 t$$

Step-by-Step Solution

Answer

The terms are \(8s\), \(2r\), and \(-7t\).

Step 1: Identify the Terms

The expression given is \(8s + 2r - 7t\). Here, we need to identify each term. In algebra, terms in an expression are the individual components separated by plus or minus signs. Thus, the terms in this expression are \(8s\), \(2r\), and \(-7t\).

Step 2: Specify Each Term

Now, let's specify each term:- The first term \(8s\) consists of a coefficient \(8\) and a variable \(s\).- The second term \(2r\) consists of a coefficient \(2\) and a variable \(r\).- The third term \(-7t\) consists of a coefficient \(-7\) and a variable \(t\). The negative sign is part of the coefficient.

Key Concepts

Terms in Algebra

In algebra, understanding terms is crucial since they form the building blocks of any algebraic expression. Essentially, a term is a single mathematical component that can comprise numbers, variables, or both combined. They are separated by plus "+" or minus "-" signs in an expression.
To simplify, think of terms as individual pieces of a puzzle that come together to form a complete picture. In the expression \(8s + 2r - 7t\):

• Each of these parts is a term: \(8s\), \(2r\), and \(-7t\).
• Notice that terms can be positive, negative, or even zero.
• The signs "\(+\)" and "\(-\)" highlight the separation and relationship between different terms.

Understanding terms help one to grasp the structure of algebraic expressions better, which is pivotal for simplifying, evaluating, and manipulating algebraic expressions.

Coefficients in Algebra

Coefficients play an essential role in understanding algebraic terms. A coefficient is the numerical factor that multiplies a variable within a term. It's important because it tells you "how much" of the variable you're dealing with.
Let’s break down the expression \(8s + 2r - 7t\):

• In the term \(8s\), "8" is the coefficient, which means "8 times the variable \(s\)".
• In \(2r\), the coefficient is "2" and implies "2 times the variable \(r\)".
• Similarly, in \(-7t\), "-7" acts as the coefficient indicating "negative 7 times the variable \(t\)".

It’s crucial to understand that the coefficient speaks volumes about the magnitude and direction (negative or positive) in relation to the variable it multiplies. This understanding aids greatly in comparing and performing operations on expressions.

Variables in Algebra

Variables form the backbone of algebra, representing unknown or general numbers in expressions and equations. They are typically denoted by letters such as \(s\), \(r\), and \(t\) in our example. Variables allow flexibility and generalization in mathematics.
Here’s how variables function in the expression \(8s + 2r - 7t\):

• In \(8s\), \(s\) is the variable, which means this term is flexible and can represent any number when needed.
• Likewise, \(r\) acts as the variable in the term \(2r\), representing an unspecified or variable quality.
• The term \(-7t\) contains the variable \(t\), again indicating that it can assume different numerical values.

Variables allow us to write formulas, create functions, and even solve problems without immediately knowing exact numerical values. Mastering variables gives students the power to explore endless mathematical possibilities.