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Chapter 11: Problem 1

Simplify each expression by combining like terms. $$4 x+3 x+6 x$$

Short Answer

Expert verified
The simplified expression is \(13x\).

Step by step solution

01

Identify Like Terms

First, we need to identify the like terms in the expression. In this case, all terms are like terms because they all involve the variable \(x\). So we have \(4x, 3x,\) and \(6x\) as like terms.
02

Combine Like Terms

Next, we add the coefficients of the like terms together. The coefficients are the numbers in front of the variable \(x\). So, add \(4 + 3 + 6\) which equals \(13\). Thus, the expression \(4x + 3x + 6x\) simplifies to \(13x\).

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Key Concepts

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

Simplifying Algebraic Expressions
When simplifying algebraic expressions, the main goal is to make the expression as concise as possible. This often involves gathering similar components, known as combining like terms. By doing this, you reduce complexity, making it more straightforward to work with.
Consider the expression provided:
  • The given expression is \(4x + 3x + 6x\).
  • Each of these terms contains the variable \(x\), which is crucial for simplification.
An expression is simplified when terms of the same kind, or like terms, are combined to form a single term. We'll explore what like terms and algebraic coefficients mean, which are fundamental to this process.
Overall, simplifying helps to clean up expressions and prepare them for further calculations.
Like Terms
Like terms are foundational in algebra when working with expressions. They are terms that have the same variable raised to the same power. This uniformity is key because it determines what terms can be combined.
In the expression \(4x + 3x + 6x\), all terms are like because they share the variable \(x\). Essentially, they are different parts of the same type of units. Think of them as similar apples in a bag.
  • The presence of the same variable, \(x\), makes them like terms.
  • No term has any additional factors other than a simple variable \(x\), keeping them uniform.
This principle allows us to easily combine them, which leads to a simpler expression. Identifying and combining like terms is critical in making expressions easier to visualize and solve.
Algebraic Coefficients
An algebraic coefficient is the numerical part of a term that multiplies the variable. This is the number you often see right in front of a variable, like the \(4\), \(3\), and \(6\) in the expression \(4x + 3x + 6x\).
Adding the coefficients together is how we combine like terms:
  • Add \(4\), \(3\), and \(6\) to get \(13\).
  • The variable \(x\) remains unchanged, since it acts as a common theme among these terms.
By performing these additions, we condense the expression into \(13x\), reflecting the total quantity of the variable \(x\) present. The coefficient tells us how many of these variables we truly have available. This highlights the significant role coefficients play in structuring and simplifying algebraic expressions.

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Most popular questions from this chapter

Translate each phrase or sentence to a mathematical expression or equation. Five more than some number is three more than four times the number.

For problems \(57-140\), solve each equation. $$ x+2=3-x $$

Translate each phrase or sentence to a mathematical expression or equation. Twice a number minus seven equals four.

For problems \(57-140\), solve each equation. $$ \frac{1 x}{2}=2 $$

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