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Chapter 3: Problem 37

Simplify each expression. $$ z-8 z $$

Short Answer

Expert verified
The simplified expression is \(-7z\).

Step by step solution

01

Identify Like Terms

We start by identifying the like terms in the expression. Here, both terms 'z' and '-8z' are like terms because they both contain the variable 'z'.
02

Combine Like Terms

Now, combine the like terms by subtracting the coefficients. The coefficient of the first term is 1 (implied), so we calculate: \( 1 - 8 = -7 \).
03

Form the Simplified Expression

Combine the result of the coefficients with the variable, so the simplified expression is \(-7z\).

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Key Concepts

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

Like Terms
In algebra, understanding like terms is an important step in simplifying expressions. Like terms are terms in an algebraic expression that have the exact same variable part, meaning they have the same variable raised to the same power.
For example, in the expression \(z - 8z\), both terms contain the variable \(z\).
Therefore, they are considered like terms.
  • Why are like terms important? They allow us to simplify expressions by combining them, reducing complexity.
  • What makes terms "like"? Having identical variables and exponents make them like terms.
Recognizing like terms can simplify complicated expressions significantly. The act of finding them is the first step in the simplification process.
Combining Coefficients
Once like terms are identified, combining coefficients is the next step. Coefficients are the numbers in front of variables in terms. In the expression \(z - 8z\), the coefficient of \(z\) is 1 (implied), and the coefficient of \(-8z\) is -8.
Combining these coefficients involves basic arithmetic.
  • Additive Combination: If terms are being added or subtracted, simply add or subtract the coefficients.
  • Example: For \(z - 8z\), we subtract: \(1 - 8 = -7\).
This process turns multiple similar terms into a single term with a new coefficient, further simplifying the expression.
Expressions with Variables
Algebra often involves dealing with expressions with variables, which are symbols representing unknown numbers. Variables allow general equations to model real-world situations or mathematical problems.
An expression is a combination of variables, numbers, and operators (like +, -, 脳, 梅). For instance, \(z - 8z\) is an expression that includes the variable \(z\).
  • Why use variables? They provide a way to deal with unknown values and form general-solving methods.
  • Relation to like terms: Variables must match for terms to be combined.
Understanding how variables work in expressions equips you for various algebraic tasks, including solving equations and simplifying expressions.

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Most popular questions from this chapter

Write each phrase as a variable expression. Use \(x\) to represent "a number." Negative ten decreased by 7 times a number

Round each number to the given place value. 52,333 to the nearest thousand

Solve each equation. \(-4(x+7)-30=3-37\)

In your own words, explain the addition property of equality.

Write each phrase as a variable expression. Use \(x\) to represent "a number." Negative eight plus a number
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