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Chapter 1: Problem 2

Simplify the algebraic expressions by combining similar terms. $$ 5 x-8 x+x $$

Short Answer

Expert verified
The simplified expression is \(-2x\).

Step by step solution

01

Identify Similar Terms

First, analyze the expression to identify similar terms. Similar terms are terms that have the same variable raised to the same power. In the expression \(5x - 8x + x\), all terms are similar because they all contain the variable \(x\).
02

Group the Similar Terms Together

Combine all the similar terms identified in the previous step. For \(5x - 8x + x\), group the coefficients of \(x\): \((5 - 8 + 1)x\).
03

Perform the Arithmetic Operations

Calculate the sum of the coefficients of \(x\) obtained in the previous step: \(5 - 8 + 1 = -2\).
04

Write the Simplified Expression

Incorporate the result from the arithmetic operation back into the expression: \(-2x\).

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Key Concepts

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

Combining Like Terms
In algebra, simplifying expressions by combining like terms is a crucial skill. Like terms share the same variables and each variable is raised to the same power. This means they can be combined through addition or subtraction.
  • Look for terms with identical variables.
  • Even if their coefficients differ, ensure the variable portions match exactly.
  • Only change the coefficients when combining like terms.
In the given example, all parts of the expression, namely \(5x\), \(-8x\), and \(x\), are like terms as they all feature the variable \(x\) raised to the first power. By combining these terms, the expression is greatly streamlined, resulting in a simpler equation to work with.
Simplification Techniques
Simplifying algebraic expressions makes them easier to interpret and work with. After identifying like terms, grouping them efficiently is the key to simplification.
  • Group the coefficients of the like terms.
  • Perform any necessary arithmetic operations.
  • Simplify the resulting expression for clarity.
Using our example, once like terms \(5x\), \(-8x\), and \(x\) are identified, their coefficients \(5\), \(-8\), and \(1\) are grouped together. Adding these coefficients, which requires simple arithmetic, gives us \(-2\). Multiply this sum by the common variable \(x\) to achieve the simplified expression \(-2x\).
Variable Coefficients
Variable coefficients represent numbers that are multiplied by variables in an algebraic expression. Understanding these is crucial for combining like terms and simplifying expressions.
  • Coefficients can be positive or negative.
  • Arithmetic operations on coefficients lead to simpler expressions.
  • The coefficient dictates the magnitude and direction of the term's value.
In the expression \(5x - 8x + x\), the coefficients are \(5\), \(-8\), and \(1\). By focusing arithmetic operations on these coefficients, we ensure the integrity of the expression while simplifying it. Thus, when these coefficients are combined to result in \(-2x\), the expression becomes more streamlined and easier to handle.

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