Problem 57 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression. $$11 a+12+3 a+2$$

Step-by-Step Solution

Answer

The simplified expression is \(14a + 14\).

Step 1: Identify Like Terms

The expression given is \(11 a+12+3 a+2\). The 'like terms' in this case are \(11a\) and \(3a\) (since they have the variable \(a\) to the power of 1) and the constants \(12\) and \(2\).

Step 2: Combine Like Terms

You 'combine' \(11a\) and \(3a\) by simply adding them together. This yields \(14a\). Likewise, you combine \(12\) and \(2\) similarly by adding. This results in \(14\).

Step 3: Rewrite the Original Expression

Having combined like terms, the original expression can be rewritten in simplified form as \(14a + 14\).

Key Concepts

Like Terms

When dealing with algebraic expressions, understanding "like terms" is crucial. Think of them as parts of the expression that can be combined because they share similar characteristics. In our example, like terms refer to those with identical variable components and exponents. This means that, if we're working with terms like \(11a\) and \(3a\), they are considered like terms. Both have the same variable, \(a\), and are raised to the same power, which is 1 in this case.

What about numbers like 12 and 2? These are constants, and they too can be categorized as like terms, simply because they're numbers without variables. Differentiating between these like terms makes simplifying expressions more manageable. Always remember, combining like terms allows us to simplify expressions by reducing them to their most basic form.

Variable Terms

Variable terms in algebra are any parts of an expression that contain a variable, like \(a\), \(b\), or \(x\). These terms can include coefficients—numbers that multiply the variable. Let's look at \(11a\) from our example. Here, 11 is the coefficient and \(a\) is the variable.

Understanding the role of variable terms is essential, especially when simplifying expressions. They show how terms relate within an equation. The coefficient expresses how many times the variable is being added to itself. This is why it's possible to add \(11a\) and \(3a\). Since the variable \(a\) is the same in both terms, we just add their coefficients. It tells us the total number of times \(a\) occurs in a side-by-side sum.

• Variable term: Any term containing a variable.
• Coefficient: A number that multiplies the variable in a term.
• Example: In \(11a\), 11 is the coefficient and \(a\) is the variable part.

Combining Like Terms

"Combining Like Terms" is the process of simplifying algebraic expressions by merging terms that are alike. It's very much like adding apples to apples and oranges to oranges.

• If you have two terms like \(11a\) and \(3a\), you're simply adding the coefficients (11 and 3) because the variable \(a\) is the same. This results in \(14a\).
• Similarly, you handle constants like 12 and 2 the same way. Adding them gives you 14.

In algebra, combining like terms is a way to "clean up" and streamline expressions, such that they are easier to handle in further calculations or equations. Doing so reduces clutter and makes it clear what quantity each variable or constant is contributing to the sum. As demonstrated in the solution, you start with an expression like \(11a + 12 + 3a + 2\) and end up with \(14a + 14\) after combining. This not only solves the problem at hand but also prepares the expression for any further operations or equations.