Problem 82 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(77-96,\) simplify each algebraic expression. $$12 x+x$$

Step-by-Step Solution

Answer

The simplified form of the equation is \(13x\).

Step 1: Identify Like Terms

In the given equation, identify like terms. Here, both \(12x\) and \(x\) are 'like' terms as they have the same variable \(x\).

Step 2: Combine Like Terms

Combine those like terms, which in this case means adding \(12x\) and \(x\) together, remembering that \(x\) stands for \(1x\) when there's no other coefficient.

Step 3: Write Down the Simplified Expression

Write down the result of combining the like terms. When you add \(12x\) and \(x\) (or \(1x\), it's the same thing), you wind up with \(13x\).

Key Concepts

Understanding Like Terms

In algebra, like terms are terms that contain the same variable raised to the same power. For example, in the expression \(12x + x\), both terms are like terms because they both contain the variable \(x\) to the first power. Only the coefficients, the numbers in front of the variable, differ. When you encounter terms like these, it signifies they can be combined to simplify the expression.
Remember, it's crucial to have the exact same variable and exponent for terms to be considered 'like'. If you encounter the expression \(3x^2 + 5x\), these are not like terms, as the variables have different exponents. Differentiating like terms from unlike terms is the first step in simplifying algebraic expressions.

Combining Like Terms

Combining like terms is just the process of adding or subtracting the coefficients of like terms. In the expression \(12x + x\), the terms both have the variable \(x\), so you can add their coefficients together.
Let's break it down:

• \(12x\) has a coefficient of 12.
• \(x\), which is the same as \(1x\), has a coefficient of 1.

Add these coefficients together: 12 plus 1 equals 13. Therefore, when you combine these like terms, you get \(13x\).
This operation is pivotal in simplifying expressions, allowing us to transform complex expressions into their simplest form.

Simplifying Expressions

Simplifying expressions involves rewriting algebraic expressions in their most concise form. This does not change the value of the expression but makes it easier to work with.
When we look at \(12x + x\), after recognizing and combining the like terms, the expression simplifies to \(13x\). The aim here is to reduce the surplus and only present the essential combined form of like terms.
Simplifying is about efficiency and clarity. A simpler expression is not only easier to read but also faster and less mistake-prone to handle in further calculations, whether you're substituting values or solving equations. Keep your expressions tidy for an easier math journey!