Problem 4 - College Algebra with Modeling and Visualization

Question

Combine like terms whenever possible. $$9 x-7 x$$

Step-by-Step Solution

Answer

The simplified expression is \(2x\).

Step 1: Identify the Terms

The given expression is made up of the terms \(9x\) and \(-7x\). Both of these terms are like terms because they each contain the variable \(x\).

Step 2: Combine Like Terms

To combine \(9x\) and \(-7x\), add or subtract their coefficients while keeping the variable \(x\). Calculate: \(9 - 7\).

Step 3: Calculate the Coefficient Difference

Subtract the coefficients: \(9 - 7 = 2\).

Step 4: Write the Final Expression

After combining like terms, the expression simplifies to \(2x\). This is the simplified expression of the given terms.

Key Concepts

Coefficient

In algebra, a coefficient is the numerical part of a term that is multiplied by the variable. For example, in the term \(9x\), the number 9 is the coefficient, showing that the variable \(x\) is being multiplied by 9. Coefficients are essential because they give magnitude and direction to the term. When dealing with coefficients, keep these points in mind:

• They can be positive or negative, indicating the direction of the term.
• They are crucial in operations such as addition and subtraction of like terms.
• They show how many times to account for the variable in an expression.

By understanding the role of coefficients, you can easily combine like terms by focusing on them. For instance, in the expression \(9x - 7x\), the coefficients 9 and -7 are combined while the variable \(x\) remains unchanged.

Simplifying Expressions

Simplifying expressions involves rewriting them in a more straightforward or reduced form. This process is essential for solving equations easily. Here are the main steps in simplifying algebraic expressions:

• Identify and combine like terms. This means merging terms that have the same variable and exponent, such as \(9x\) and \(-7x\).
• Perform arithmetic operations on coefficients. For our exercise, this meant subtracting 7 from 9, giving us the simplified result of \(2x\).
• Write the final expression clearly, showcasing your simplified term with its coefficient.

Simplification helps streamline complex problems, making it easier to understand and solve further operations. By breaking down expressions into simpler parts, you can manage them effectively.

Variables

Variables are symbols used to represent numbers or values in expressions and equations. Typically, variables in algebra are denoted by letters, such as \(x\) in our example. They are placeholders that can take on various values depending on the context of the equation.
Variables are fundamental to understand because:

• They allow expressions to represent general situations, not fixed values. This makes algebra flexible and wide-ranging in applications.
• They interact with coefficients, where changes in the coefficient affect the term's value without changing the variable itself.
• They enable the process of forming equations, helping convey relationships between different quantities easily.

Understanding the role of variables helps when simplifying expressions and working through algebraic equations as they consistently appear in various mathematical problems.