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Chapter 2: Problem 41

Simplify each expression. See Examples 3 through 7 . $$ -9 x+4 x+18-10 x $$

Short Answer

Expert verified
The expression simplifies to \(-15x + 18\).

Step by step solution

01

Combine Like Terms in Variables

First, identify and combine all the like terms involving the variable \( x \) in the expression. The given terms in the expression are \(-9x\), \(4x\), and \(-10x\).
02

Simplify Variable Expression

Add the coefficients of \( x \): \(-9 + 4 - 10 = -15\). Thus, combining the \( x \) terms gives \(-15x\).
03

Identify Constant Terms

Identify any constant terms in the expression; here, it is the number 18. There are no other constants to combine with 18.
04

Combine Results

Combine the simplified \( x \) terms with the constant term. Therefore, the expression simplifies to \(-15x + 18\).

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Key Concepts

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

Like Terms
When working with algebraic expressions, identifying like terms is a crucial step. Like terms are terms that have the exact same variables raised to the same power. For instance, in our exercise, - the terms - \(-9x\), - \(4x\), and - \(-10x\)are like terms because they all involve the variable \(x\) raised to the first power.To simplify expressions, we combine these like terms by adding or subtracting their coefficients, making it easier to manage the expression. Like terms allow us to systematically shorten our expressions by reducing the number of operations and terms involved in calculations.
Variables and Coefficients
Every algebraic term is made up of two main components: variables and coefficients.The variable is the letter that represents a number we don't know yet, while the coefficient is the number in front of the variable showing how many times the variable is being counted. In the expression \(-9x + 4x - 10x +18\):- The variable is \(x\), shared across multiple terms. - The coefficients are - \(-9\) in \(-9x\), - \(4\) in \(4x\), and - \(-10\) in \(-10x\).When simplifying expressions, we primarily focus on the coefficients to merge all the like terms together, effectively reducing the overall complexity of the expression.
Expression Simplification
Simplifying an expression is all about making it shorter or easier to work with while ensuring the expression's original value or meaning is maintained. To simplify an expression like \(-9x + 4x - 10x +18\), start by identifying and combining like terms. Here are some simple steps to follow:
  • Identify all like terms, as done in the example with the \(x\) terms.
  • Add or subtract their coefficients to combine them into a single term.
  • Keep track of constant terms, like 18 in this expression, and ensure they are not altered incorrectly when combining variables.
The result of simplification is often a cleaner, easier-to-handle expression like \(-15x + 18\). This is much simpler to solve in equations or use in further algebraic manipulation.

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