Problem 27 - Algebra A Combined Function

Question

Simplify each expression by combining like terms. See Examples 6 through 10. $$ 9 x-20 x $$

Step-by-Step Solution

Answer

The simplified expression is \(-11x\).

Step 1: Identify the Like Terms

In the expression \( 9x - 20x \), both terms are like terms because they both contain the variable \( x \). Like terms have the same variable raised to the same power. Here, both terms have the same variable \( x^1 \).

Step 2: Combine the Coefficients

To combine like terms, add their coefficients together. The expression \( 9x - 20x \) can be combined by subtracting the coefficients: \( 9 - 20 \).

Step 3: Perform the Arithmetic Operation

Calculate the result of subtracting 20 from 9. This yields \( 9 - 20 = -11 \).

Step 4: Simplified Expression

Attach the result from the arithmetic operation to the variable. Therefore, \( 9x - 20x = -11x \).

Key Concepts

Like Terms

In expressions, like terms are terms that have the same variables raised to the same powers. This is a crucial concept when simplifying expressions as it allows us to combine them. For instance, in the expression \(9x - 20x\), both terms have the variable \(x\) raised to the power of 1. Therefore, they are considered like terms.

• Like terms share the same variable.
• They also have the same exponent.

This means we can simplify the expression by adding or subtracting the coefficients of these terms. Understanding like terms helps you efficiently combine and reduce expressions.

Coefficients

Coefficients are the numerical parts of terms in an expression. In the expression \(9x - 20x\), the numbers 9 and -20 are the coefficients. They indicate how many times the variable is being multiplied by a number.

• Coefficients can be positive or negative numbers.
• They do not include the variable part.

To combine like terms, we manipulate these coefficients via arithmetic operations. It's important to remember that coefficients strictly refer to the numbers connected to variables, not standalone numbers in an expression.

Variables

Variables are symbols used in expressions to represent unknown or changing numbers. Typically, letters like \(x\), \(y\), and \(z\) are used. In our example, \(x\) is the variable involved, providing a placeholder that can vary.

• Variables are often letters of the alphabet.
• They can represent numbers in equations and functions.

When you encounter a variable in an expression, it's crucial to pay attention to the coefficients attached to the variable since they determine the magnitude of the terms involved in the arithmetic operations.

Arithmetic Operations

Arithmetic operations refer to basic mathematical operations such as addition, subtraction, multiplication, and division. In simplifying expressions, these operations are used to combine like terms.

• Addition and subtraction are generally involved in simplifying like terms.
• The results of these operations affect the coefficients of the terms.

In \(9x - 20x\), subtraction is used to simplify the expression. When operations are correctly performed, the expression \(9 - 20 = -11\) shows that the terms are simplified to \(-11x\). Arithmetic operations are essential for obtaining a simplified, equivalent expression.