Problem 48 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 3 z-6 z+8 z $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression: \(3z-6z+8z\) Answer: \(5z\)

Step 1: Identify the like terms

In the expression \(3z-6z+8z\), all terms have the same variable 'z' raised to the power of 1.

Step 2: Combine like terms

Add or subtract the coefficients of the like terms: \((3-6+8)z\)

Step 3: Calculate the combined coefficient

Perform the calculation inside the parenthesis: \((3-6+8)= 5\)

Step 4: Write the simplified expression

Replace the coefficients in the expression with the calculated combined coefficient: \(5z\). So, the simplified algebraic expression is \(5z\).

Key Concepts

Understanding Like Terms

When working with algebraic expressions, identifying like terms is a critical initial step. Like terms are terms that have the same variable parts. For example, in the expression \(3z - 6z + 8z\), each term is a like term because they all contain the variable \(z\).
Without same-like terms, you cannot combine them directly. All you need to look for in like terms is a common variable and the same exponent. Whether positive or negative, terms must share these attributes to be combined.

Combining Coefficients

After identifying the like terms, the next step is to focus on their coefficients—these are the numbers in front of the variables. To combine like terms, you only need to add or subtract their coefficients.
In our practice expression \(3z - 6z + 8z\), the coefficients that you need to combine are 3, -6, and 8.

• First, manage these like numeric values, taking their signs into account.
• The process: perform the operations \((3 - 6 + 8)\) to find the new coefficient.
• The result of this operation will dictate the magnitude of the single term in your simplified expression.

By focusing on combining coefficients, you simplify the expression down to its essentials.

The Art of Simplification

Simplification in algebra is about translating a complex expression into an easily understandable form. Once you have processed the coefficients of like terms, replacing them with a singular term represents this concept beautifully.
In simplest terms, simplification means making an expression easier to manage.

• Calculate the combined coefficient like we did: \((3 - 6 + 8) = 5\).
• Attach the resultant coefficient to the shared variable to finish: \(5z\).

This makes an expression immediately recognizable and easier to use in future calculations or discussions.

The Role of Variables

Variables are the foundational elements in algebraic expressions. They stand in for unknown or changeable numbers, allowing expressions to remain flexible and applicable in numerous situations.
In our example \(3z - 6z + 8z\), 'z' is the variable. No matter the coefficients, the presence of 'z' allows us to represent a potentially infinite set of possibilities.

• Variables allow you to explore relationships and changes between numbers.
• The goal in expressions is often to simplify these, reducing them to the simplest form with a focus on the role and impact of the variable.

Recognizing variables is essential, as they tie together the entire expression, no matter how much you simplify the coefficients.