Problem 34 - 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 simplified expression, and they are 3x and 7.

Step 1: Combine like terms

First, we need to identify and combine the like terms in the expression. In this case, we can combine the terms with 'x': 2x and x. $$ 2x + (x+7) = 2x + x + 7 $$

Step 2: Simplify the expression

Now, let's add the terms together: 2x and x, to simplify the expression. $$ 2x + x + 7 = (2 + 1)x + 7 = 3x + 7 $$

Step 3: Count the terms and list them

Finally, count the number of terms and list them out. In this simplified expression, we have two terms, 3x and 7. Number of terms: 2 Terms: 3x, 7

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, operators (like addition or multiplication), variables (like x, y, or z), and sometimes exponents (like ^2 which stands for squaring a number or variable). For example, 2x + (x + 7) is an algebraic expression where x is the variable. These expressions are the backbone of algebra and are used to describe patterns and to formulate relationships between quantities.

When working with algebraic expressions, it's essential to be able to identify parts of the expression, such as terms, coefficients, and constants. A term is a single mathematical entity that can be a number, a variable, or the product of numbers and variables. Coefficients are the numerical part of the terms that involve variables, and constants are terms that are just numbers.

Understanding the structure of algebraic expressions helps us to manipulate them in order to simplify expressions or solve equations.

Simplifying Expressions

Simplifying an algebraic expression means reducing it to its simplest form. This often involves combining like terms, which are terms in the expression that have the exact same variable part, such as in our example 2x + (x + 7). Like terms can be combined because they represent the same quantities, much like how you can combine apples with apples, but not apples with oranges.

To simplify the expression 2x + x + 7, we combine the like terms (2x and x) to get 3x + 7, making the simplified expression easier to work with in further mathematical calculations. The simplified form better reveals the core components of the expression and is essential when solving equations or performing operations such as substitution.

The process of simplifying expressions is foundational to algebra and is a frequent requirement across different areas of mathematics. Simplified expressions are easier to read, understand, and utilize in more complex procedures.

Terms in Algebra

Terms in algebra are the distinct elements of an algebraic expression that are typically separated by plus (+) or minus (-) signs. Each term can be a number (constant), a variable, or a combination of both (e.g., 3x or -2y^2). When presented with an expression like 3x + 7, we can easily identify that 3x and 7 are the two terms we're dealing with.

Knowing the number of terms in an expression is key to understanding its structure. This is why, when faced with simplifying algebraic expressions, the first thing we often do is combine like terms to reduce the expression to as few terms as possible. For instance, 2x + (x + 7) initially appears to have three terms, but after combining like terms, we can see that it only has two: 3x and 7.

In summary, being able to recognize and work with terms is essential in algebra. It allows us to dissect complex expressions into more manageable pieces, which we can then simplify, evaluate, or manipulate according to the problem we are trying to solve.