Problem 2 - College Algebra with Modeling and Visualization

Question

Combine like terms whenever possible. $$-9 z+6 z$$

Step-by-Step Solution

Answer

-3z

Step 1: Identify Like Terms

Like terms are terms that have the same variable and exponent. In the expression \(-9z + 6z\), both terms are like terms because they both have the variable \(z\).

Step 2: Combine Coefficients

To combine like terms, add or subtract the coefficients (the numerical parts) of the terms. Here, you subtract 9 from 6.Calculate: \(6 - 9 = -3\)

Step 3: Write the Combined Term

After combining the coefficients, write the result along with the variable. So,\( -9z + 6z = -3z \)

Key Concepts

Combining Like Terms

When we talk about combining like terms in algebra, we mean simplifying expressions by merging terms that have the same variables raised to the same power. This is possible because those terms are directly comparable and can be added or subtracted from one another.
For example, in the expression \(-9z + 6z\), both terms are like terms as they each have the variable \(z\). Because they share this commonality, we can proceed to combine them by operating on their coefficients. In this case, the operation is subtraction because of the negative sign in front of \(9z\).

• Identify like terms: Same variables and exponents
• Combine them by adding or subtracting coefficients

Combining like terms makes the expression simpler and more manageable, which is often necessary for solving algebraic equations.

Linear Expressions

Linear expressions are algebraic statements where variables are raised to the power of one and are not multiplied times each other. They are made up of terms like \(ax + b\), where \(a\) and \(b\) are constants.
A key feature of linear expressions is their simplicity, allowing them to be easily graphed as straight lines on a coordinate plane. The original exercise, \(-9z + 6z\), is a simple example of a linear expression because the variable \(z\) is raised to the first power.

• Simplify by combining like terms
• Shorten and neaten an expression

Understanding linear expressions and their properties aids in formulating equations and solutions efficiently.

Coefficients

Coefficients are the numerical parts of terms in an expression. They are directly in front of the variables and dictate how many copies of the variable you have. In the expression \(-9z + 6z\), \(-9\) and \(6\) are the coefficients.
Combining like terms relies heavily on manipulating coefficients. To merge like terms, you simply add or subtract these numbers while keeping the variable part unchanged. In the example problem, subtract \(9\) from \(6\) to get \(-3\). The resulting expression is \(-3z\).

• Numbers multiplying a variable
• Determine the magnitude and direction of the variable

Grasping the role of coefficients helps you see how terms influence equations and model real-world situations.