Problem 80 - Elementary and Intermediate Algebra

Question

Simplify by combining like terms. $$ 9 z-7-z-19 z $$

Step-by-Step Solution

Answer

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

Step 1: Identify Like Terms

Identify the terms in the expression that are like terms. Remember, like terms are terms that have the exact same variables raised to the same power. In this expression, \(9z\), \(-z\), and \(-19z\) are all like terms because they all contain the variable \(z\).

Step 2: Combine Like Terms

Add or subtract the coefficients of the like terms to combine them. The expression \(9z - z - 19z\) consists of the terms \(9z\), \(-1z\), and \(-19z\). Combine them by calculating \(9 + (-1) + (-19) = -11\).

Step 3: Write the Simplified Expression

Write down the expression using the simplified coefficient from Step 2. This gives us \(-11z\).

Key Concepts

Understanding Like Terms

To truly grasp the concept of like terms, it's important to understand that they are terms in an algebraic expression that contain the same variable raised to the same power. A simple way to think about it is to see if the terms have identical variable parts. If they do, then you have identified like terms. For example, in the expression \(9z - 7 - z - 19z\), the terms \(9z\), \(-z\), and \(-19z\) all contain the variable \(z\) with the power of 1.

The term \(-7\) does not have a variable at all, so it stands alone and is not ‘like’ the others. Identifying like terms is crucial because it allows you to simplify expressions more easily.

The Art of Combining Like Terms

Once you've identified like terms, the next step is to combine them. Combining like terms involves adding or subtracting their coefficients, which are the numbers directly in front of the variables. This process simplifies the expression, making it easier to work with.

In our example \(9z - z - 19z\), you notice each term involves \(z\). When combining them, you’ll add or subtract only the coefficients:

• Start with the first coefficient \(9\).
• Then deal with \(-1\) (because \(-z\) is the same as \(-1z\)).
• Finally, handle \(-19\).

So, perform the calculation: \(9 + (-1) + (-19) = -11\). Thus, \(-11z\) is the resulting term after combining like terms.

This process allows you to rewrite the expression in its simplest form.

Exploring Variable Coefficients

In mathematics, variable coefficients are the numbers that are multiplied by the variables in an expression. They indicate how many times a variable is being added together. Understanding coefficients is essential in simplifying algebraic expressions.

Let's break it down with an example: Consider the term \(-19z\). Here, \(z\) is the variable, and \(-19\) is the coefficient. This implies that the term represents \(-19\) times \(z\).

By manipulating and rearranging variable coefficients, you simplify expressions and solve equations more efficiently.

• For positive coefficients, you just add them up.
• For negative coefficients, you subtract them.

This skill is vital for solving algebraic expressions as it helps in balancing and simplifying equations. Ultimately, understanding coefficients allows for more fluid manipulation of algebraic expressions.