Problem 17 - Prealgebra

Question

Identify the terms, like terms, coefficients, and constants in each expression. \(5 c-2 d+3 d-d\)

Step-by-Step Solution

Answer

Terms: \(5c, -2d, +3d, -d\); Like terms: \(-2d, +3d, -d\); Coefficients: \(5, -2, 3, -1\); Constants: None.

Step 1: Identify the Terms

To identify the terms in the expression, look at the distinct groups separated by '+', or '-'. The given expression is \(5c - 2d + 3d - d\). The terms in this expression are \(5c\), \(-2d\), \(+3d\), and \(-d\).

Step 2: Group Like Terms

Like terms are terms that have the same variable raised to the same power. In the expression \(5c - 2d + 3d - d\), the like terms are \(-2d\), \(+3d\), and \(-d\) because they all contain the variable \(d\). The term \(5c\) is not like any other term because it has a different variable.

Step 3: Identify the Coefficients

Coefficients are numerical factors in front of variables in each term. In \(5c\), the coefficient is \(5\); in \(-2d\), the coefficient is \(-2\); in \(+3d\), the coefficient is \(+3\); in \(-d\), the coefficient is \(-1\) (since \(d\) can be rewritten as \(1d\)).

Step 4: Identify Constants

Constants are terms that do not contain a variable. In this expression, there are no constants because all terms include variables.

Key Concepts

Terms Identification

In algebra, an expression is made up of terms. Terms are the individual parts of the expression that are separated by addition or subtraction signs. For example, in the expression \(5c - 2d + 3d - d\), you can identify the terms by looking at each distinct part between the '+' and '-' signs. The terms here are \(5c\), \(-2d\), \(+3d\), and \(-d\).
Each of these terms consists of a number multiplied by a variable, or sometimes just a standalone number, but in this expression, each comes with a variable component.

Like Terms

Like terms are those terms which have the same variable raised to the same power. This means they can be combined together. Let's take a closer look at the expression \(5c - 2d + 3d - d\). Here, the like terms are \(-2d\), \(+3d\), and \(-d\). All these terms share the same variable, \(d\), and can be added or subtracted from one another.
Although \(5c\) is also a term, it is not like any of the other terms because it contains a different variable \(c\). Identifying like terms is essential for simplifying algebraic expressions and solving equations.

Coefficients

The coefficient in a term is the numerical factor that multiplies the variable within the term. In other words, it's the number in front of the variable. Taking the example expression \(5c - 2d + 3d - d\):

• The term \(5c\) has a coefficient of \(5\).
• The term \(-2d\) has a coefficient of \(-2\).
• The term \(+3d\) has a coefficient of \(+3\).
• The term \(-d\) can be rewritten as \(-1d\), so its coefficient is \(-1\).

This makes coefficients extremely important because they impact the value represented by the variable in an algebraic term.

Constants

In algebraic expressions, constants are the terms without variables. They maintain a fixed value and do not change. However, examining the expression \(5c - 2d + 3d - d\), you will notice that there are no constants present.
Every term in this expression involves a variable, either \(c\) or \(d\). It illustrates that not all expressions have constants, but when they do, they are typically standalone numbers that can be counted on to remain the same regardless of other variables in an expression. Understanding the role of constants aids in simplifying expressions and solving equations.