Problem 63 - Elementary and Intermediate Algebra

Question

List the like terms in each expression, if any. $$ 3 x+2-2 x $$

Step-by-Step Solution

Answer

Like terms are \(3x\) and \(-2x\).

Step 1: Identify the Terms

The expression given is \(3x + 2 - 2x\). First, let's list the individual terms: \(3x\), \(2\), and \(-2x\). These are the components we will examine for like terms.

Step 2: Identify Like Terms

Like terms are terms that have the same variable raised to the same power. In this case, both \(3x\) and \(-2x\) have the variable \(x\) raised to the same power (which is 1), so they are like terms. The constant term \(2\) does not have a variable and cannot be combined with \(3x\) or \(-2x\).

Step 3: List the Like Terms

Based on the identification from Step 2, the like terms in the expression \(3x + 2 - 2x\) are \(3x\) and \(-2x\). The constant term \(2\) stands alone as it does not have the same variable component as the other terms.

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operations. It is like a sentence in a mathematical language. These expressions can range from a simple combination of numbers and variables to more complex combinations involving various operations such as addition, subtraction, multiplication, and division.

• For example, in the expression \(3x + 2 - 2x\), we have numbers and variables combined through addition and subtraction.
• Each part separated by a plus or minus sign is called a term. In this case, the terms are \(3x\), \(2\), and \(-2x\).

Understanding how to manipulate these expressions is crucial in algebra, as it allows us to solve equations and simplify formulas.

Combining Like Terms

Combining like terms is a key step in simplifying algebraic expressions. Like terms are terms that contain the same variables raised to the same power. These terms can be combined by adding or subtracting their coefficients.

• For example, in the expression \(3x + 2 - 2x\), we identify that \(3x\) and \(-2x\) are like terms because they both contain the variable \(x\).
• The constant \(2\) does not have a variable, so it stands alone and is not combined with the other terms.

To simplify, combine the coefficients of the like terms: \(3x - 2x\) results in \((3-2)x = 1x\) which simplifies to \(x\). Hence, the expression \(3x + 2 - 2x\) simplifies to \(x + 2\). No change is made to the constant, as it has no counterparts.

Variables and Constants

Understanding variables and constants is fundamental in algebra.

• Variables, often denoted by letters such as \(x\), represent unknown numbers that can change. They are the "mystery" in an equation or expression.
• Constants are numbers without any variables attached. They are fixed values that do not change regardless of the variables in the expression. In the expression \(3x + 2 - 2x\), the number \(2\) is a constant.

By identifying and understanding these components, you can better analyze and simplify algebraic expressions. Recognizing which parts of an expression are constants and which are variables is the starting point for solving algebraic equations and combining like terms effectively.