Problem 8 - Intermediate Algebra

Question

Fill in the blanks. Simplifying the sum or difference of like terms is called ______ like terms.

Step-by-Step Solution

Answer

The process is called 'combining' like terms.

Step 1: Identify Like Terms

Like terms are terms that have the same variable raised to the same power. For example, in the expression \(3x + 5x\), both terms are like terms because they both have the variable \(x\).

Step 2: Define the Process

Identifying like terms allows you to add or subtract their coefficients. The process of combining these coefficients is necessary for simplifying an expression.

Step 3: Fill in the Blank

The process of simplifying an expression by adding or subtracting the coefficients of like terms is called 'combining.' So, the blank should be filled with 'combining'.

Key Concepts

Like Terms

In algebra, a critical concept is understanding what like terms are. Like terms are terms within an algebraic expression that have the same variable parts raised to the same power. Recognizing like terms is essential because it allows you to perform operations such as addition and subtraction on these terms. For example, take the expression \(2x^2 + 3x - x + 4x^2\):

• Here, \(2x^2\) and \(4x^2\) are like terms because they both contain the variable \(x\) raised to the second power.
• Similarly, \(3x\) and \(-x\) are like terms as they share the variable \(x\) raised to the first power.

Identifying like terms is a foundational step in simplifying algebraic expressions as it enables you to efficiently combine them.

Simplifying Expressions

The process of simplifying expressions involves combining like terms to create a simpler form of the expression. Simplifying expressions is an essential skill in algebra, as it makes complex problems more manageable and the results more straightforward. Once you have identified like terms, the next step is to perform the arithmetic operation needed - either addition or subtraction of the coefficients - of those terms. For example:
Let's simplify the expression \(5y - 3 + 7y + 2\).

• First, identify the like terms: \(5y\) and \(7y\) are like terms, and \(-3\) and \(+2\) are constants.
• Combine the like terms: \(5y + 7y = 12y\); thus, the expression becomes \(12y - 1\) after calculating \(-3 + 2 = -1\).

This simplified expression, \(12y - 1\), is much easier to understand and work with in further calculations.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operational symbols such as addition and subtraction. Algebraic expressions are the building blocks of algebra, allowing us to model and solve real-world problems. To work with algebraic expressions efficiently, one must be familiar with various concepts such as variables, coefficients, and terms. For instance, in the expression \(4a + 6 - 3b\):

• \(4a\) is a term with a variable \(a\), and its coefficient is 4.
• \(6\) is a constant term.
• \(-3b\) is another term with a variable \(b\), and its coefficient is -3.

Understanding how these parts of an algebraic expression interact is crucial for performing operations like addition, subtraction, and simplification. Simplifying expressions by combining like terms is a necessary step in ensuring that algebraic equations can be resolved more easily and accurately.