Problem 36 - Elementary Algebra

Question

For the expressions in the following problems, write the number of terms that appear and then list the terms. $$ a+1+(a-1) $$

Step-by-Step Solution

Answer

Answer: The total number of terms in the simplified expression is 1.

Step 1: Identify terms

To identify terms in the given expression $$a + 1 + (a - 1)$$, we start by removing the parentheses.

Step 2: Remove parentheses

Remove the parentheses, which gives us the expression: $$ a + 1 + a - 1 $$

Step 3: Combine like terms

Next, we combine like terms (if possible): - We have two 'a' terms: $$a + a$$. - We have two constant terms: $$1 - 1$$.

Step 4: Simplify the expression

Now, we simplify the expression by adding the like terms: $$ 2a + 0 $$ The expression simplifies to: $$ 2a $$

Step 5: Count and list the terms

In the simplified expression $$2a$$, we have: - Number of terms: 1 - Terms: $$2a$$

Key Concepts

Combining Like Terms

Understanding how to combine like terms is a fundamental skill in simplifying algebraic expressions. The process involves adding or subtracting variables and constants that are 'like,' meaning they have the same variable part. For instance, in the expression \(a + 1 + (a - 1)\), we observe two types of terms: the variable term \(a\) and the constant terms \(1\) and \(\−1\).

To effectively combine like terms, we first remove any grouping symbols such as parentheses. In our example, this gives us \(a + 1 + a - 1\). You'll then group like terms together which, for our case, are \(a + a\) and \(1 - 1\). This action sets the stage for simplification, by adding or subtracting these like terms to arrive at a more streamlined expression, such as \(2a\). Simplifying algebraic expressions makes them easier to work with in subsequent mathematical operations.

Simplifying Expressions

The goal of simplifying expressions in algebra is to rewrite them in their simplest form, which makes them more understandable and easier to use in further calculations. Simplification could involve combining like terms, as mentioned previously, as well as applying the distributive property, factoring, and canceling common factors, among other operations.

Let's take our expression \(a + 1 + a - 1\). After combining like terms, we get \(2a\). The constant terms \(1 - 1\) cancel each other out, leaving us with \(0\), which does not affect the value of an expression when added, hence it is omitted. The simplified expression is \(2a\), which is not only more concise but has also revealed that there is just one term present in the expression.

Algebraic Terms

Algebraic terms are the building blocks of algebraic expressions. An algebraic term could be a constant like \(5\), a variable like \(x\), or a combination of both, such as \(3x^2\). Each term is separated by a plus \(+\) or minus \(−\) sign within an expression. When we identify and count the terms in an expression such as \(2a\), we are looking for these individual parts.

In our example, the simplified expression \(2a\) consists of one term - \(2a\), even though we started with multiple terms in the original expression \(a + 1 + (a - 1)\). Recognizing these algebraic terms is critical for applying algebraic procedures accurately and for understanding the structure of algebraic expressions.