Problem 21 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 3 a+5 a+2 a $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression: \(3a + 5a + 2a\). Answer: \(10a\)

Step 1: Identify like terms

In the expression \(3a + 5a + 2a\), all the terms have the same variable 'a'. These terms are considered like terms and can be combined.

Step 2: Add coefficients of like terms

To simplify the expression, add the coefficients (the numbers in front of the variable) of the like terms: \(3 + 5 + 2 = 10\).

Step 3: Write the simplified expression

Combine the sum of the coefficients with the variable 'a': \(10a\). The simplified expression is \(10a\).

Key Concepts

Like Terms

Understanding the concept of 'like terms' is essential when simplifying algebraic expressions. In an expression, like terms are terms that have exactly the same variable(s) raised to the same power. In other words, they're the apples with the apples and the oranges with the oranges. For example, in the expression \(3a + 5a + 2a\), all the terms are like terms because they all contain the single variable 'a' raised to the first power.

When identifying like terms, it's important to note that \(3a\) and \(3a^{2}\) are NOT like terms, because the variable 'a' is raised to different powers. However, \(3a\) and \(2a\) would be like terms. To recognize them quickly, look for the matching variable parts and ignore the coefficients for a moment; if what's left matches across different terms, you've found like terms!

Coefficients

Coefficients in algebraic expressions are the numerical factors that multiply the variables. Think of them as the 'multipliers' in an expression. For instance, in \(3a\), the number 3 is the coefficient, showing how many times 'a' is being considered. Coefficients can be positive or negative numbers and are crucial when it comes to combining like terms.

When simplifying expressions such as \(3a + 5a + 2a\), focus on adding or subtracting the coefficients only. Pretend for a moment that the variable doesn't exist; you're simply working with the numbers in front. Since algebra follows the same arithmetic rules you're already familiar with, you just need to add or subtract these coefficients as ordinary numbers to simplify the expression.

Combining Like Terms

Combining like terms is the process of simplifying an algebraic expression by adding or subtracting the coefficients of terms that have the same variable part. It's like consolidating items in a warehouse; you'd put all similar items together to get a clearer view of what you have in total.

Let's use the expression \(3a + 5a + 2a\) as an example. Here's how to combine the like terms, in simple steps:

• Identify the like terms: All instances of 'a' are like terms.
• Add the coefficients: Add the numbers in front of the like terms (3, 5, and 2).
• Place the total coefficient back in front of the variable: The sum 3 + 5 + 2 gives 10, so you rewrite the combined term as \(10a\).

By combining like terms, you're effectively 'cleaning up' the expression and making it easier to work with.Keep in mind that you can only combine like terms; attempting to combine unlike terms (such as 'a' and 'b', or 'a' and 'a^2') would be like trying to add cats and dogs - it just doesn't work!