Problem 91 - Algebra A Combined Function

Question

Simplify. \(5 a+6 a\)

Step-by-Step Solution

Answer

The simplified expression is \(11a\).

Step 1: Identify Like Terms

Examine the expression \(5a + 6a\). Both terms have the variable \(a\) and can be added together as they are like terms, meaning they have the same variable raised to the same power.

Step 2: Add the Coefficients

Since the terms are like terms, you can add their coefficients (the numbers in front of the variable). The coefficient for the first term is 5, and for the second term it is 6. Add these coefficients together: \(5 + 6 = 11\).

Step 3: Attach the Variable

After adding the coefficients, attach the common variable \(a\) to the result of the addition: \(11a\).

Key Concepts

Like Terms

In algebra, **like terms** are terms that have the same variable parts. This means each term has the same letters or variables, and these variables are raised to the same power. For instance, in the expression \(5a + 6a\), both terms are considered like terms because they each have the variable \(a\). Like terms can be combined through addition or subtraction because their variable parts are identical.

• For example, \(3b + 4b\) are like terms because they both include the variable \(b\).
• Terms such as \(2x^2 + 3x\) are not like terms because the exponents on \(x\) are different.

Recognizing like terms is fundamental when simplifying algebraic expressions. Anytime you see like terms, think about combining them by adding or subtracting the coefficients.

Coefficients

A **coefficient** is the numerical part of a term, often seen directly in front of a variable. For instance, in the term \(7x\), the 7 is the coefficient. Coefficients indicate how many times the term's variable is counted. Knowing the coefficients helps you to combine or manipulate terms effectively.

• In \(5a\), the coefficient is 5.
• In \(8x^2\), the coefficient is 8.

During the process of simplifying expressions with like terms, add or subtract the coefficients while keeping the variable parts the same. For example, when simplifying \(5a + 6a\), you add the coefficients 5 and 6 to get 11, retaining the shared variable \(a\) to form the term \(11a\). This process is crucial to ensure that the simplified expression accurately represents the original terms.

Simplifying Expressions

**Simplifying expressions** is the process of making an algebraic expression easier to read or work with by reducing it to its simplest form. This often involves combining like terms and reducing the expression as much as possible. Simplifying helps in solving equations more efficiently by making them less cluttered.
Steps for simplifying:

• Identify all like terms in the expression.
• Combine the coefficients of like terms.
• Rewrite the expression with these combined terms.

For instance, to simplify \(5a + 6a\), first recognize both parts have the variable \(a\), making them like terms. Combine by adding their coefficients, 5 and 6, to produce the simplified expression \(11a\). Always remember, simplifying expressions is a valuable skill as it allows for clearer insight into the nature of algebraic relationships and prepares the groundwork for more advanced manipulations. By practicing simplifying, you'll improve your ability to handle more complex algebraic equations confidently.