Problem 47 - Prealgebra

Question

Use the distributive property to combine similar terms. \(3 a+a\)

Step-by-Step Solution

Answer

The expression simplifies to \(4a\).

Step 1: Identify Like Terms

In the expression \(3a + a\), both terms involve the variable \(a\). This means they are like terms and can be combined.

Step 2: Apply the Distributive Property

The distributive property allows us to rearrange and combine terms with the same variable. In this case, \(3a + a\) can be rewritten using the distributive property as \((3+1)a\).

Step 3: Simplify the Coefficients

Add together the coefficients of the terms: \(3 + 1\), which equals \(4\).

Step 4: Write the Simplified Expression

After adding the coefficients, the simplified expression is \(4a\).

Key Concepts

Understanding Like Terms

Like terms are terms in an expression that have the same variable raised to the same power. For instance, in the expression \(3a + a\), both terms are considered like terms because they share the variable \(a\). Recognizing like terms is the first step in simplifying expressions as it allows us to efficiently combine them. When terms are alike, you only combine the coefficients—the numbers in front of the variables.

• Identify terms with the same variable and exponent.
• Ensure all coefficients are numerical and can be easily added or subtracted.

Identifying like terms helps streamline the process of solving or simplifying algebraic expressions, making complex problems more manageable.

Combining Like Terms

Once you've identified like terms, the next step is to combine them. This process involves adding or subtracting the coefficients while retaining the common variable. In the given expression \(3a + a\), both terms share the variable \(a\), allowing us to combine them.Here's how this works:

• Write the expression as a sum of like terms: \(3a + 1a\).
• Add the coefficients: \(3 + 1\). This results in \(4\).

Combining like terms turns a complicated expression into a simpler one. This is a crucial step in many algebraic processes and helps you get to the solution more quickly and efficiently. Remember, the variable remains unchanged—you're only manipulating the numbers!

Simplifying Expressions with the Distributive Property

Simplifying expressions involves reducing them to their most compact and comprehensible form. The distributive property is a valuable tool in this process, especially when dealing with expressions involving like terms.In our example, \(3a + a\), applying the distributive property enables us to rewrite it as \((3 + 1)a\). This makes adding the coefficients straightforward. Simplifying the expression in this manner provides clarity and often leads directly to the final solution.Steps to simplify using the distributive property:

• Group like terms together. In this case, both terms with \(a\).
• Add the coefficients inside the parentheses: \(3 + 1\).
• Multiply the result by the common variable to get \(4a\).

This process not only makes expressions easier to work with but also lays the groundwork for more advanced algebraic manipulations. Mastering these concepts is essential for efficiently tackling algebra problems.