Problem 31 - Algebra for College Students

Question

Is \(f(x)=(3 x-2)-(2 x+1)\) a linear function? Explain your answer.

Step-by-Step Solution

Answer

Yes, the function \(f(x) = x - 3\) is linear.

Step 1: Identify the Components

To determine if the function \(f(x) = (3x - 2) - (2x + 1)\) is linear, start by recognizing that linear functions generally have the form \(ax + b\), where \(a\) and \(b\) are constants. Here, we need to first examine what the operations within the parentheses represent.

Step 2: Perform the Subtraction

Simplify the expression by performing subtraction between the two provided expressions. Distribute the subtraction to terms within the parentheses: \[(3x - 2) - (2x + 1) = 3x - 2 - 2x - 1.\]

Step 3: Combine Like Terms

Combine the terms with the variable \(x\) and the constant terms separately. From the previous step: \[3x - 2x = x,\] and \[-2 - 1 = -3.\] This simplifies the expression to \(f(x) = x - 3\).

Step 4: Check for Linear Form

Now that the expression is simplified to \(f(x) = x - 3\), it matches the linear form \(f(x) = ax + b\), where \(a = 1\) and \(b = -3\). This confirms that \(f(x)\) is a linear function.

Key Concepts

Expression Simplification

Expression simplification is a key step in solving many algebraic problems. It involves making a complex expression easier to understand or work with by performing operations. For example, in the function \(f(x) = (3x - 2) - (2x + 1)\), we need to simplify this expression by performing operations between the terms inside the parentheses. To simplify, distribute the operations such as multiplication or subtraction to all terms within the parentheses. Here, we subtract the second expression from the first one. The subtraction is applied to every term within the parentheses:

• The expression becomes: \[3x - 2 - 2x - 1\]

This distribution helps break down the problem into more manageable parts, which is crucial for combining like terms and further evaluating the function.

Combining Like Terms

Combining like terms is an important process in algebra that helps in simplifying expressions by reducing the number of terms. Like terms are terms that have the same variables raised to the same power. Once we distribute terms in an expression like in the example \(3x - 2 - 2x - 1\), we need to group and simplify terms:

• Combine all the terms with the variable \(x\): \(3x - 2x = x\)
• Next, combine the constant terms: \(-2 - 1 = -3\)

After combining like terms, the simplified expression is \(f(x) = x - 3\). This step is critical in making the expression more straightforward, revealing its true form and helping us analyze it further.

Linear Equation

A linear equation is an equation that can be written in the form \(ax + b\), where \(a\) and \(b\) are constants, and \(x\) is the variable. The graph of a linear equation is a straight line. Once the expression is simplified and like terms are combined, analyzing the structure of \(f(x) = x - 3\) reveals that it conforms to the linear form \(ax + b\):

• Here, \(a = 1\) and \(b = -3\).
• Since both \(a\) and \(b\) are constants, \(f(x)\) is a linear function.

Understanding linear equations is important because they model scenarios where there is a constant rate of change, making them foundational in algebra and applicable to real-world situations like budgeting or predicting trends.