Problem 6 - Elementary and Intermediate Algebra

Question

When we write \(9 x+x\) as \(10 x,\) we say we have ______ like terms.

Step-by-Step Solution

Answer

Combined

Step 1: Identify the Terms

First, look at the expression provided: \(9x + x\). These terms both include the variable \(x\). Because they have the same variable raised to the same power, they are considered 'like terms'.

Step 2: Combine Like Terms

To combine \(9x + x\), add the coefficients together. The coefficients are the numbers in front of the variable \(x\), which are 9 and 1 respectively. So, we add these coefficients: \(9 + 1 = 10\).

Step 3: Write the Combined Expression

After adding the coefficients, attach the common variable \(x\) to get the expression \(10x\). This process is known as combining like terms.

Step 4: Fill in the Blank

The exercise involves filling in the blank with the process performed. When we write \(9x + x\) as \(10x\), we have **combined** like terms.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They do not have an equality sign, unlike equations. Think of them as sentences written with numbers and symbols. For instance, in the expression \(9x + x\), you have two terms each involving a coefficient and a variable part.

However, algebraic expressions can vary in complexity and length. They can be as simple as \(3x\) or quite complex with multiple terms and operations, like \(3x^2 - 2x + 7\). When you deal with algebraic expressions, your main task often involves simplifying them by reducing their terms or solving them for a given variable. Simplification is frequently accomplished by the process of combining like terms.

Like Terms

When working with algebraic expressions, you'll often encounter the concept of like terms. But what exactly are like terms?

• Simply put, like terms are terms that have the exact same variable part. This means both the variable and its exponent must be identical.
• In the expression \(9x + x\), both terms include the variable \(x\) raised to the power of 1. Therefore, they are categorized as like terms.

Identifying like terms is crucial because it allows you to simplify expressions by combining them. For example, \(4y\) and \(5y\) are like terms, while \(4y\) and \(5x\) are not because they involve different variables. The key takeaway is that like terms are essential for simplifying expressions and making them easier to work with.

Coefficient Addition

Once you've identified the like terms in an algebraic expression, the next step is often combining them. This is where coefficient addition comes into play.

"Coefficients" are the numerical parts of the terms. They are the numbers placed in front of the variables. For instance, in \(9x + x\), the coefficients are 9 and 1 respectively. To combine these terms, simply add their coefficients: \(9 + 1 = 10\). Following this, attach the common variable \(x\) to complete the simplification, resulting in \(10x\).

• It’s essential to remember that you can only add (or subtract) the coefficients of like terms — terms with the same variable.
• Through coefficient addition, you achieve a simpler and often more understandable expression.

Mastering coefficient addition is foundational in solving and simplifying various algebraic expressions proficiently.