Problem 1 - Fundamentals of Mathematics

Question

For problems \(1-10\), specify each term. $$ 6 a-2 b+5 c $$

Step-by-Step Solution

Answer

The terms are \(6a\), \(-2b\), and \(5c\).

Step 1: Identify the expression

The given expression is \(6a - 2b + 5c\). This is an algebraic expression composed of three different terms.

Step 2: Recognize individual terms

Each term in the expression is a distinct part, usually separated by a '+' or '-' sign. Here, there are three terms: \(6a\), \(-2b\), and \(5c\).

Step 3: Specify each term

- \(6a\) is the first term, where 6 is the coefficient and \(a\) is the variable.- \(-2b\) is the second term, where -2 is the coefficient and \(b\) is the variable.- \(5c\) is the third term, where 5 is the coefficient and \(c\) is the variable.

Step 4: Conclude the specification

Each term has been specified with its respective coefficient and variable: \(6a\), \(-2b\), \(5c\).

Key Concepts

Terms Identification

In algebra, identifying terms is crucial for understanding and manipulating expressions. A term is a single mathematical expression that can be a number, a variable, or numbers and variables multiplied together. Terms are usually separated by either a plus or a minus sign. This makes them relatively easy to spot within an algebraic expression. For instance, in the expression \(6a - 2b + 5c\), we identify the terms by looking between the '+' and '-' signs.

• First Term: \(6a\)
• Second Term: \(-2b\)
• Third Term: \(5c\)

Each of these represents a different part of the expression that can stand on its own when evaluated for specific variable values. Understanding how to identify terms helps in rewriting and solving algebraic expressions efficiently.

Coefficients in Algebra

The coefficient in an algebraic term is the numerical factor that multiplies the variable. It provides information about the magnitude or size of that term in the context of the expression. In our example, each term has a distinct coefficient which can be a positive or negative number.

• For the term \(6a\), the coefficient is 6.
• For the term \(-2b\), the coefficient is -2.
• For the term \(5c\), the coefficient is 5.

Coefficients are vital because they tell us how much of the variable is present in each term. They also play a critical role when we perform operations like addition, subtraction, or factorization of algebraic expressions. Positive coefficients indicate addition, while negative coefficients imply subtraction, impacting how terms are combined.

Variables in Expressions

Variables are symbols that represent unknown values or can vary within mathematical expressions. They are fundamental elements of algebra because they allow us to model real-world situations that are dynamic and not fixed. In the expression \(6a - 2b + 5c\), each term has its variable.

• In the term \(6a\), the variable is \(a\).
• In the term \(-2b\), the variable is \(b\).
• In the term \(5c\), the variable is \(c\).

Substituting different numbers for these variables allows us to evaluate the expression for various scenarios. The versatility of variables is that they make expressions flexible and broadly applicable across different problems. Understanding variables helps us form equations and solve problems beyond static numerical calculations.