Problem 45 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(-6 x-3 x\)

Step-by-Step Solution

Answer

The simplified expression is \( -9x \).

Step 1: Identify Like Terms

Determine that \( -6x \) and \( -3x \) are like terms since they both contain the variable \( x \) to the same power.

Step 2: Combine Coefficients

Add the coefficients of the like terms together: \( -6 + (-3) \).

Step 3: Simplify the Sum

Simplify the sum of the coefficients: \( -6 + (-3) = -9 \).

Step 4: Rewrite the Expression

Combine the simplified coefficient with the variable: \( -9x \).

Key Concepts

like terms

In algebra, 'like terms' are terms that have the same variable raised to the same power. Identifying like terms is crucial for simplifying expressions. For example, in the exercise \( -6x - 3x \), both terms contain the variable \( x \) and are raised to the first power. Since they are like terms, they can be combined. This step lays the foundation for further simplification by allowing us to handle terms with the same variable together.

combining coefficients

Once we've identified the like terms, the next step is 'combining coefficients'. The coefficient is the numerical part of the term that multiplies the variable. In the term \( -6x \), \( -6 \) is the coefficient, while in \( -3x \), \( -3 \) is the coefficient. To combine these coefficients, we add them together: \[ -6 + (-3) = -9 \]. Understanding how to properly add positive and negative numbers is key to this step.

algebraic simplification

'Algebraic simplification' involves rewriting expressions in a simpler form. In our exercise, after combining the coefficients, we simplify the expression \( -6x - 3x \) to \( -9x \). This process makes expressions easier to work with and understand. The goal is to reduce the expression to its simplest form without changing its value. Simplification helps in solving equations and understanding the relationship between variables.

variable manipulation

Variable manipulation is a fundamental aspect of algebra. It involves performing operations on variables while following algebraic rules. In the given exercise, the variable \( x \) remains unchanged during the simplification process. The manipulation happens with the coefficients, where \( -6x \) and \( -3x \) combine to form \( -9x \). Effective variable manipulation skills make solving complex equations and expressions more manageable.