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Chapter 3: Problem 51

Simplify each expression. $$ 8-x+4 x-2-9 x $$

Short Answer

Expert verified
The simplified expression is \(-6x + 6\).

Step by step solution

01

Identify Like Terms

First, we need to identify like terms in the expression. Like terms are terms that have the same variable raised to the same power. In this expression, the like terms are \(x\) terms: \(-x\), \(4x\), and \(-9x\), and the constant terms: \(8\) and \(-2\).
02

Combine Like Terms with x

Combine the like terms with the variable \(x\): \(-x + 4x - 9x\). To simplify, perform the arithmetic operation on the coefficients of \(x\). This gives us \(-1 + 4 - 9 = -6\), so the \(x\) terms simplify to \(-6x\).
03

Combine Constant Terms

Next, combine the constant terms: \(8 - 2\). Perform the subtraction to find the simplified constant terms: \(6\).
04

Write the Simplified Expression

Put together the results from Steps 2 and 3. The simplified expression is \(-6x + 6\).

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Key Concepts

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

Like Terms
To simplify an algebraic expression, the first step is to identify like terms. Like terms are those that contain the same variables raised to the same power. For example, in the expression given,
  • The terms \(-x\), \(4x\), and \(-9x\) are like terms because they all contain the variable \(x\).
  • Similarly, the numbers without variables, such as \(8\) and \(-2\), are considered constant terms or like terms among themselves.
It is important to recognize these like terms because the next step in simplifying the expression involves combining them.
Combining Terms
Once you've identified the like terms, you can start combining them. This means you simply add or subtract the coefficients (the numbers in front of the variables) of the like terms. For example, to combine the terms with \(x\) in the given expression:
  • The terms \(-x\), \(4x\), and \(-9x\) get combined by adding their coefficients: \(-1 + 4 - 9\), which simplifies to \(-6\).
  • Similarly, combine the constant terms \(8\) and \(-2\) by performing the operation: \(8 - 2 = 6\).
By combining like terms this way, you simplify the expression into its most streamlined form.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols (like \(+\) or \(-\)). In essence, it is a combination of terms.For the expression \(8 - x + 4x - 2 - 9x\):
  • The numbers \(8\) and \(-2\) are the constant terms.
  • The terms \(-x\), \(4x\), and \(-9x\) include the variable \(x\).
Algebra often involves simplifying such expressions by combining like terms to reach a more concise form, such as the simplified result \(-6x + 6\). This process of simplifying expressions is crucial in solving equations and understanding mathematical relationships.

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