Problem 20 - Fundamentals of Mathematics

Question

Specify each term. $$7 a-2 b-3 c-4 d$$

Step-by-Step Solution

Answer

The terms are \(7a\), \(-2b\), \(-3c\), and \(-4d\).

Step 1: Identify the Terms

In an algebraic expression, terms are separated by plus (+) or minus (−) signs. In the expression \(7a - 2b - 3c - 4d\), the terms are \(7a\), \(-2b\), \(-3c\), and \(-4d\).

Step 2: Specify Each Term

Each term in an expression consists of a numerical coefficient and a variable part. Identify each term:- \(7a\) has a coefficient of \(7\) and a variable part \(a\).- \(-2b\) has a coefficient of \(-2\) and a variable part \(b\).- \(-3c\) has a coefficient of \(-3\) and a variable part \(c\).- \(-4d\) has a coefficient of \(-4\) and a variable part \(d\).

Key Concepts

Coefficients

Coefficients are an essential component of algebraic expressions. In simpler terms, they are the numbers that are placed in front of variables. They tell you how many times to multiply that variable. For instance, in the term \(7a\), the number 7 is the coefficient. It indicates that you have 7 units of the variable \(a\). Similarly, in the expression \(-2b\), \(-2\) is the coefficient, illustrating a negative association with the variable \(b\). Coefficients can be positive or negative integers, fractions, or decimals.

• Positive Coefficient: Suggests a positive relationship with the variable. Example: In \(4x\), 4 is the coefficient.
• Negative Coefficient: Indicates a negative relationship, as seen in \(-3y\), where \(-3\) is the coefficient.

Understanding coefficients helps you recognize the quantity and direction in which a variable is "tied" in the expression. They play a crucial role when solving equations.

Terms in Algebra

An algebraic expression is made up of terms. Each term is a part of the expression that is separated by a plus (+) or minus (−) sign. For instance, in the expression \(7a - 2b - 3c - 4d\), there are four distinct terms: \(7a\), \(-2b\), \(-3c\), and \(-4d\). Each term consists of a coefficient and a variable part.

• Single Term: An expression like \(5x\) is a single term as it stands alone.
• Multiple Terms: An expression with more, such as \(3x + 4y\), has two terms (\(3x\) and \(4y\)).

Breaking down an expression into its terms allows you to manage each part individually, which is particularly helpful in operations such as addition, subtraction, or even factoring.

Variables

Variables are symbols used to represent unknown quantities in mathematical expressions. They are essential for forming expressions, equations, and inequalities. In our earlier expression \(7a - 2b - 3c - 4d\), the letters \(a\), \(b\), \(c\), and \(d\) are the variables.

• Common Variable Letters: Typically, letters like \(x\), \(y\), and \(z\) are used, but any letter can serve as a variable.
• Role of Variables: They allow expressions to be generalized, enabling the solution of equations across different scenarios.
• Substitution: Variables can be replaced by specific numbers depending on the given problem context, which helps in solving equations.

Understanding variables is central to learning algebra, as they introduce the concept of unknowns in mathematical problems, providing the foundation for more complex operations and problem-solving techniques.