Problem 25 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ a+8 a+3 a $$

Step-by-Step Solution

Answer

Answer: The simplified form of the given algebraic expression is $$12a$$.

Step 1: Identify the like terms

In the expression, we have three terms with the same variable 'a': $$ a + 8a + 3a $$

Step 2: Add the coefficients of the like terms

We need to add the coefficients of each term. The coefficient of 'a' is 1, the coefficient of '8a' is 8, and the coefficient of '3a' is 3. Adding these coefficients, we get: $$ 1 + 8 + 3 $$

Step 3: Calculate the sum of the coefficients

Adding the coefficients, we obtain: $$ 1 + 8 + 3 = 12 $$

Step 4: Write the simplified expression

Now that we have the sum of the coefficients, we multiply the result by the common variable 'a' to get the simplified expression: $$ 12a $$ The simplified expression is: $$12a$$.

Key Concepts

Like Terms

In algebra, like terms are terms that have the same variable raised to the same power. Recognizing like terms is crucial because it allows us to simplify expressions by combining them.
For example, in the expression \(a + 8a + 3a\), each term contains the variable 'a'. Therefore, these terms are like terms. Their coefficients can be added together to simplify the expression.
Here are key points to remember about like terms:

• Same variable: The terms must contain the same variable (e.g., 'a').
• Same exponent: The variable should be raised to the same power.
• Can be combined: You can add or subtract their coefficients.

Mastering the identification of like terms makes the process of simplifying expressions more manageable.

Coefficients

A coefficient is the numerical factor of a term containing a variable. It tells you how many times the variable is multiplied. For instance, in the term \(8a\), 8 is the coefficient.
Understanding coefficients is essential because they determine how like terms can be combined.
In the expression \(a + 8a + 3a\), the coefficients are 1, 8, and 3 respectively (it's a good habit to remember that there is an implicit 1 as the coefficient of \(a\)).
Here are important points about coefficients:

• They explain the quantity: Coefficients specify how many of the variable you have.
• They can be combined: When you add or subtract like terms, you are combining the coefficients.
• Invisible ones: If a variable stands alone, it has an implicit coefficient of 1.

By adding these coefficients, the power of simplification becomes evident as seen when we combine \(1 + 8 + 3\).

Simplification

Simplification is the process of reducing an algebraic expression into its simplest form. It involves combining like terms by adding or subtracting their coefficients, and thus making expressions more manageable.
Let's take \(a + 8a + 3a\):

• Identify like terms: Recognize that all terms have the variable 'a'.
• Add the coefficients: Combine the numbers in front of the like terms which are 1, 8, and 3.
• Multiply by the common variable: The sum of the coefficients is 12, hence the simplified expression becomes \(12a\).

The result, \(12a\), is much simpler and easier to interpret. Being able to simplify expressions accurately is an important skill in algebra that helps in both solving equations and understanding polynomial expressions.