Problem 107 - Algebra A Combined Function

Question

Simplify each expression by combining any like terms. $$ 3 x-5 x+7 $$

Step-by-Step Solution

Answer

The simplified expression is \(-2x + 7\).

Step 1: Identify Like Terms

First, identify the terms in the expression that can be combined. The expression is \(3x - 5x + 7\). The like terms here are \(3x\) and \(-5x\) since they both involve the variable \(x\).

Step 2: Combine Like Terms

Add the coefficients of the like terms \(3x\) and \(-5x\) together. This means you calculate \(3 - 5\), which gives \(-2\). So, the expression becomes \(-2x + 7\).

Step 3: Constant Term

Notice that the constant term \(+7\) does not have any like terms, so it stays as it is in the simplified expression.

Key Concepts

Simplification of Expressions

Simplifying expressions involves rewriting them in a more concise and manageable form without changing their value. The main goal is to make expressions easier to work with, especially when solving equations or inequalities. In the exercise, the expression \(3x - 5x + 7\) is simplified by combining terms that are similar in nature, known as "like terms".When simplifying, follow these steps:

• First, look for terms that share the same variables and exponents. These are your potential like terms that can be combined.
• Then, perform operations like addition or subtraction on the coefficients of these like terms.
• Ensure that any constant terms—which are terms without variables—are also considered in the final expression.

Simplifying expressions can make solving mathematical problems quicker and less error-prone. It helps you see the relationships between numbers and variables more clearly.

Coefficients

Coefficients are numerical factors that multiply the variable in an expression. Understanding coefficients is crucial when combining like terms.In the expression \(3x - 5x + 7\), the coefficients of the terms involving \(x\) are 3 and -5, respectively. These numbers tell us how many units of the variable \(x\) we have and are key players in simplifying expressions.When you combine like terms, you primarily deal with the coefficients:

• Add or subtract the coefficients depending on whether they are being added or subtracted in the expression.
• In this exercise, calculating \(3 - 5\) results in the new coefficient, \(-2\), for the \(x\) term.

Coefficients can be positive, negative, or even zero, and they play a crucial role in the form and value of the final expression. Paying close attention to the sign (positive or negative) of each coefficient is important to avoid mistakes.

Like Terms

Like terms are terms in an expression that have the same variables raised to the same power. Recognizing like terms is an essential skill because it allows you to combine and simplify expressions effectively.In this exercise, \(3x\) and \(-5x\) are like terms:

• Both terms contain the variable \(x\) to the first power.
• This commonality allows us to combine them by only operating on their coefficients.

When combining like terms, one must:

• Focus on the coefficients and perform the same operation indicated in the expression (addition, subtraction, etc.).
• Keep any term with a unique variable or constant as is, unless another like term can be found.

In the context of simplifying the expression, finding like terms helps to streamline and reduce the number of terms, allowing for easier manipulation and interpretation of mathematical problems.