Problem 33 - Elementary Algebra

Question

For the expressions in the following problems, write the number of terms that appear and then list the terms. $$ 2 x+x+7 $$

Step-by-Step Solution

Answer

Answer: There are 2 terms in the given expression, which are $$3x$$ and $$7$$.

Step 1: Combine like terms

We can see that there are like terms in the given expression. Specifically, \(2x\) and \(x\) are like terms. Let's combine them to simplify the expression. $$ 2x + x + 7 = (2 + 1)x + 7 = 3x + 7 $$

Step 2: Count the number of terms

In the simplified expression, we now have two terms: \(3x\) and \(7\). So, the number of terms is 2.

Step 3: List the terms

We will now list the terms in the expression. The terms are as follows: 1. \(3x\) 2. \(7\) Therefore, the given expression has 2 terms, which are \(3x\) and \(7\).

Key Concepts

Combining Like Terms

Combining like terms is an essential skill in simplifying algebraic expressions. It's akin to organizing your shopping list - grouping all the apples together and all the oranges together. In mathematics, like terms are terms that have the same variable raised to the same power. For instance, in \(2x + x + 7\), both \(2x\) and \(x\) are like terms because each term contains the variable \(x\).
To combine them, you simply add the coefficients of the like terms together. A coefficient is the number in front of the variable. So, for \(2x\) and \(x\), you add the coefficient 2 with the understood coefficient 1 (as \(x\) is the same as \(1x\)). Hence, \(2x + x\) becomes \(3x\).

• Identify the like terms in the expression.
• Add the coefficients of these like terms.
• Rewrite the expression using the combined terms.

Simplifying Expressions

Simplifying expressions involves making an expression as concise and recognizable as possible. After combining like terms, you continue by arranging the expression in its simplest form.
This makes it easier to understand and use in calculations. In our example, once we've turned \(2x + x + 7\) into \(3x + 7\), we've simplified it. The expression \(3x + 7\) is much easier to work with!
When simplifying:

• Look for any similar terms that can be combined.
• Ensure that the expression is organized, often starting with variables followed by constant numbers.
• Check that no more adjustments can be made to keep the expression "simple."

Counting Terms

Counting terms in an expression is like counting the ingredients for a recipe - you need to know each component to understand what you're working with. In an algebraic expression, terms are separated by addition or subtraction signs.
After simplifying \(2x + x + 7\) to \(3x + 7\), we see there are two distinct parts to the expression. These are the terms!

• Identify the terms by seeing what's separated by '+' or '-'.
• Count each distinct item that is not further reducible into simpler elements.

In this simplified expression, the terms are \(3x\) and \(7\), making the total number of terms two. Understanding how to count terms helps in solving equations and assessing the complexity of algebraic expressions.