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Chapter 2: Problem 23

Simplify each expression, when possible. SEE EXAMPLE 1. (OB]ECTIVE 1) $$8 x+12 x$$

Short Answer

Expert verified
The simplified expression is \(20x\).

Step by step solution

01

Identify Like Terms

In the expression \(8x + 12x\), the like terms are \(8x\) and \(12x\). Here 'like terms' refer to terms that share the same variable to the same power. In this case, both terms are multiples of the variable \(x\).
02

Combine Like Terms

To combine like terms, you can add or subtract the numerical coefficients. In this case, you'll add the coefficients \(8\) and \(12\). This gives you \(8 + 12 = 20\).
03

Write Final Solution

Combining the like terms, you end up with the simplified expression of \(20x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
Simplifying expressions is a fundamental skill in algebra that involves reducing expressions to their simplest form. This process often involves combining similar mathematical components. When you simplify, you aim to make an expression easier to understand and work with, which is particularly useful in solving equations or performing various arithmetic operations.
The core steps usually involve:
  • Identifying like terms within the expression
  • Combining these like terms
  • Rewriting the expression in its simplest form
By following these steps, you can transform complex expressions into simpler ones, making future calculations much more manageable.
Like Terms
The concept of like terms is essential when it comes to simplifying expressions. Like terms are terms in an expression that have the same variables raised to the same powers. They can have different coefficients, which are numbers in front of the variables, but as long as the variable part is identical, they are considered like terms.
For instance:
  • Terms like \(8x\) and \(12x\) are like terms because they both have the variable \(x\).
  • Terms like \(5y^2\) and \(7y^2\) are also like terms because they share the variable and exponent \(y^2\).
Combining like terms is straightforward, as it involves adding or subtracting their coefficients while keeping the variable part unchanged. Understanding this concept is key to mastering algebraic manipulation.
Variables and Coefficients
Variables and coefficients form the backbone of algebraic expressions. A variable is a symbol, typically a letter, used to represent an unknown number. The variable in our example expression, \(8x + 12x\), is \(x\).
Coefficients are the numerical parts of the terms and they are situated directly in front of variables. In the expression \(8x + 12x\), the coefficients are \(8\) and \(12\). These numbers determine how many times the variable is counted in that term.
Here鈥檚 a simple breakdown:
  • The variable dictates what kind of terms are considered like terms
  • The coefficient is the part that you add or subtract when combining like terms
Getting familiar with these components will greatly aid in simplifying expressions and understanding more complex algebraic functions.

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