Problem 49 Simplify each expression. $$ ... [FREE SOLUTION]

Chapter 0: Problem 49

Simplify each expression. $$ 2+3(2 z-5)-3(4 z+6)-8 $$

Short Answer

Expert verified

-6z - 39

Step by step solution

Distribute the Constants

Distribute the constants inside the parentheses. For the term $3(2z-5)$, distribute 3: \[ 3 \times 2z = 6z \] and \[ 3 \times -5 = -15 \]. For the term $-3(4z+6)$, distribute -3: \[ -3 \times 4z = -12z \] and \[ -3 \times 6 = -18 \]. So, the expression becomes: \[ 2 + 6z - 15 - 12z - 18 - 8 \].

Combine Like Terms

Combine the like terms. The expression is now: \[ 2 + 6z - 15 - 12z - 18 - 8 \]. Combine the constants: \[ 2 - 15 - 18 - 8 = -39 \]. Combine the z terms: \[ 6z - 12z = -6z \]. So, the simplified expression is: \[ -6z - 39 \].

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Key Concepts

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

distributive property

The Distributive Property is a fundamental tool in algebra. It allows you to multiply a single term by each term inside a set of parentheses. This property is expressed as $ a(b+c) = ab + ac $. In the exercise, we applied this property to both $ 3(2z - 5) $ and $ -3(4z + 6) $.

Here's what happens in detail:

For $ 3(2z - 5) $, multiply 3 by both $ 2z $ and $ -5 $:
$ 3 \times 2z = 6z $ and $ 3 \times -5 = -15 $.
For $ -3(4z + 6) $, multiply -3 by both $ 4z $ and $ 6 $:
$ -3 \times 4z = -12z $ and $ -3 \times 6 = -18 $.

By distributing the constants correctly, you break down a complex expression into simpler parts, making it easier to simplify further.

combining like terms

Combining Like Terms is a crucial step when simplifying algebraic expressions. Terms are considered 'like' if they have the same variables raised to the same power. In our solution, we combined constants with constants and terms involving $z$ with each other.

The steps are:

List all like terms together: our expression was $ 2 + 6z - 15 - 12z - 18 - 8 $.
Combine the constants:

$ 2 - 15 = -13 $
$ -13 - 18 = -31 $
$ -31 - 8 = -39 $

Combine the $z$ terms:

$ 6z - 12z = -6z $

This gives us the simplified expression $ -6z - 39 $. Understanding how to combine like terms effectively is significant in simplifying complex algebraic expressions.

algebra for college students

Algebra serves as the foundation to many advanced mathematical concepts. For college students, mastering algebra is essential not only for math courses but for subjects like physics, engineering, and economics. The two main concepts covered in our exercise鈥擠istributive Property and Combining Like Terms鈥攑rovide tools for tackling more complex problems in these fields.

Here鈥檚 why these concepts matter:

Distributive Property: Helps in simplifying expressions and equations, which is crucial for solving real-world problems.
Combining Like Terms: Assists in making expressions simpler and clearer, allowing for easier manipulation and solution of equations.

Applying these foundational skills can improve problem-solving abilities and mathematical thinking, which are valuable in various academic and professional contexts.

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91影视

Simplify each expression. $$ 2+3(2 z-5)-3(4 z+6)-8 $$

Short Answer

Step by step solution

Distribute the Constants

Combine Like Terms

Key Concepts

distributive property

combining like terms

algebra for college students

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