Select All Equations Equivalent To $-18x + 54$.A. $-6(3x + 9$\]B. $-18(-x - 3$\]C. $18(-x + 3$\]D. $-6(3x - 9$\]

Introduction

In mathematics, equivalent equations are expressions that have the same value or solution. Identifying equivalent equations is a crucial skill in algebra, as it allows us to simplify complex expressions and solve problems more efficiently. In this article, we will explore the concept of equivalent equations and provide a step-by-step guide on how to select equivalent equations.

What are Equivalent Equations?

Equivalent equations are expressions that have the same value or solution. They can be obtained by multiplying or dividing both sides of an equation by the same non-zero value. For example, the equations $2x = 6$ and $x = 3$ are equivalent because they have the same solution.

How to Select Equivalent Equations

To select equivalent equations, we need to follow these steps:

1. Distribute the negative sign: When a negative sign is distributed to a term inside parentheses, it changes the sign of the term. For example, $-6(3x + 9)$ becomes $-18x - 54$.
2. Distribute the negative sign to the terms inside the parentheses: When a negative sign is distributed to the terms inside the parentheses, it changes the sign of each term. For example, $-18(-x - 3)$ becomes $18x + 54$.
3. Distribute the negative sign to the terms inside the parentheses and change the sign of each term: When a negative sign is distributed to the terms inside the parentheses and changes the sign of each term, it becomes $-18x + 54$.
4. Distribute the negative sign to the terms inside the parentheses and change the sign of each term, and then multiply by -1: When a negative sign is distributed to the terms inside the parentheses and changes the sign of each term, and then multiplied by -1, it becomes $18x - 54$.

Selecting Equivalent Equations: A Step-by-Step Guide

Now that we have understood the concept of equivalent equations and the steps to select them, let's apply this knowledge to the given problem.

Step 1: Distribute the Negative Sign

The first option is $-6(3x + 9)$. To distribute the negative sign, we need to change the sign of each term inside the parentheses.

-6(3x + 9) = -6(3x) - 6(9)
= -18x - 54

Step 2: Distribute the Negative Sign to the Terms Inside the Parentheses

The second option is $-18(-x - 3)$. To distribute the negative sign, we need to change the sign of each term inside the parentheses.

-18(-x - 3) = -18(-x) - 18(-3)
= 18x + 54

Step 3: Distribute the Negative Sign to the Terms Inside the Parentheses and Change the Sign of Each Term

The third option is $18(-x + 3)$. To distribute the negative sign, we need to change the sign of each term inside the parentheses.

18(-x + 3) = 18(-x) + 18(3)
= -18x + 54

Step 4: Distribute the Negative Sign to the Terms Inside the Parentheses and Change the Sign of Each Term, and Then Multiply by -1

The fourth option is $-6(3x - 9)$. To distribute the negative sign, we need to change the sign of each term inside the parentheses.

-6(3x - 9) = -6(3x) + 6(9)
= -18x + 54

Conclusion

In conclusion, the correct answer is $-6(3x + 9)$, $-18(-x - 3)$, $-6(3x - 9)$, and $18(-x + 3)$ are all equivalent to $-18x + 54$. These equations can be obtained by distributing the negative sign to the terms inside the parentheses and changing the sign of each term.

Final Answer

The final answer is:

• A. $-6(3x + 9)$
• B. $-18(-x - 3)$
• C. $18(-x + 3)$
• D. $-6(3x - 9)$
  Frequently Asked Questions: Selecting Equivalent Equations ===========================================================

Q: What is the difference between equivalent equations and equivalent expressions?

A: Equivalent equations and equivalent expressions are both mathematical expressions that have the same value or solution. However, equivalent equations are typically used to describe the relationship between two or more variables, while equivalent expressions are used to describe a single value or expression.

Q: How do I determine if two equations are equivalent?

A: To determine if two equations are equivalent, you can use the following steps:

1. Check if the equations have the same variables: If the equations have the same variables, but with different coefficients or constants, they may be equivalent.
2. Check if the equations have the same structure: If the equations have the same structure, but with different coefficients or constants, they may be equivalent.
3. Check if the equations can be transformed into each other: If the equations can be transformed into each other by multiplying or dividing both sides by the same non-zero value, they are equivalent.

Q: What is the importance of selecting equivalent equations?

A: Selecting equivalent equations is an important skill in mathematics because it allows us to:

1. Simplify complex expressions: By selecting equivalent equations, we can simplify complex expressions and make them easier to work with.
2. Solve problems more efficiently: By selecting equivalent equations, we can solve problems more efficiently and accurately.
3. Understand mathematical concepts: By selecting equivalent equations, we can gain a deeper understanding of mathematical concepts and relationships.

Q: How do I select equivalent equations in a real-world scenario?

A: In a real-world scenario, you may need to select equivalent equations to:

1. Model real-world problems: By selecting equivalent equations, you can model real-world problems and make predictions about the behavior of complex systems.
2. Analyze data: By selecting equivalent equations, you can analyze data and make informed decisions about business or personal matters.
3. Solve optimization problems: By selecting equivalent equations, you can solve optimization problems and find the best solution to a given problem.

Q: What are some common mistakes to avoid when selecting equivalent equations?

A: Some common mistakes to avoid when selecting equivalent equations include:

1. Not checking if the equations have the same variables: Make sure that the equations have the same variables before trying to select equivalent equations.
2. Not checking if the equations have the same structure: Make sure that the equations have the same structure before trying to select equivalent equations.
3. Not checking if the equations can be transformed into each other: Make sure that the equations can be transformed into each other by multiplying or dividing both sides by the same non-zero value.

Q: How do I practice selecting equivalent equations?

A: To practice selecting equivalent equations, you can:

1. Work on math problems: Practice working on math problems that involve selecting equivalent equations.
2. Use online resources: Use online resources, such as math websites or apps, to practice selecting equivalent equations.
3. Join a study group: Join a study group or find a study partner to practice selecting equivalent equations together.

Conclusion

In conclusion, selecting equivalent equations is an important skill in mathematics that can be used to simplify complex expressions, solve problems more efficiently, and understand mathematical concepts. By following the steps outlined in this article and practicing regularly, you can become proficient in selecting equivalent equations and apply this skill to real-world scenarios.