What Is The Justification For The Step Taken From Line 2 To Line 3?$\[ \begin{array}{c} 3x + 9 - 7x = X + 10 + X \\ -4x + 9 = 2x + 10 \\ -6x + 9 = 10 \\ -6x = 1 \\ x = \frac{1}{6} \end{array} \\]A. The Subtraction Property Of Equality B. The

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What is the Justification for the Step Taken from Line 2 to Line 3?

Understanding the Algebraic Manipulation

In the given algebraic manipulation, we are asked to justify the step taken from line 2 to line 3. To do this, we need to understand the properties of equality and how they are applied in algebraic manipulations.

The Properties of Equality

There are three main properties of equality that are used in algebraic manipulations:

  • The Addition Property of Equality: This property states that if two expressions are equal, then adding the same value to both expressions will result in two new expressions that are also equal.
  • The Subtraction Property of Equality: This property states that if two expressions are equal, then subtracting the same value from both expressions will result in two new expressions that are also equal.
  • The Multiplication Property of Equality: This property states that if two expressions are equal, then multiplying both expressions by the same non-zero value will result in two new expressions that are also equal.
  • The Division Property of Equality: This property states that if two expressions are equal, then dividing both expressions by the same non-zero value will result in two new expressions that are also equal.

The Step from Line 2 to Line 3

The step from line 2 to line 3 is:

-4x + 9 = 2x + 10 → -6x + 9 = 10

To justify this step, we need to apply the Subtraction Property of Equality. This property states that if two expressions are equal, then subtracting the same value from both expressions will result in two new expressions that are also equal.

In this case, we subtract 2x from both sides of the equation -4x + 9 = 2x + 10. This results in:

-4x + 9 - 2x = 2x + 10 - 2x

Simplifying the equation, we get:

-6x + 9 = 10

Therefore, the step from line 2 to line 3 is justified by the Subtraction Property of Equality.

Conclusion

In conclusion, the step taken from line 2 to line 3 is justified by the Subtraction Property of Equality. This property is used to subtract the same value from both sides of an equation, resulting in two new expressions that are also equal.

The Algebraic Manipulation

Now that we have justified the step from line 2 to line 3, let's take a closer look at the entire algebraic manipulation.

The given algebraic manipulation is:

3x + 9 - 7x = x + 10 + x -4x + 9 = 2x + 10 -6x + 9 = 10 -6x = 1 x = 1/6

Let's break down each step of the algebraic manipulation and justify it using the properties of equality.

Step 1: 3x + 9 - 7x = x + 10 + x

To justify this step, we need to apply the Subtraction Property of Equality. This property states that if two expressions are equal, then subtracting the same value from both expressions will result in two new expressions that are also equal.

In this case, we subtract 7x from both sides of the equation 3x + 9 - 7x = x + 10 + x. This results in:

3x - 7x + 9 = x + 10 + x - 7x

Simplifying the equation, we get:

-4x + 9 = 2x + 10

Therefore, the step from line 1 to line 2 is justified by the Subtraction Property of Equality.

Step 2: -4x + 9 = 2x + 10

To justify this step, we need to apply the Subtraction Property of Equality. This property states that if two expressions are equal, then subtracting the same value from both expressions will result in two new expressions that are also equal.

In this case, we subtract 2x from both sides of the equation -4x + 9 = 2x + 10. This results in:

-4x + 9 - 2x = 2x + 10 - 2x

Simplifying the equation, we get:

-6x + 9 = 10

Therefore, the step from line 2 to line 3 is justified by the Subtraction Property of Equality.

Step 3: -6x + 9 = 10

To justify this step, we need to apply the Subtraction Property of Equality. This property states that if two expressions are equal, then subtracting the same value from both expressions will result in two new expressions that are also equal.

In this case, we subtract 9 from both sides of the equation -6x + 9 = 10. This results in:

-6x + 9 - 9 = 10 - 9

Simplifying the equation, we get:

-6x = 1

Therefore, the step from line 3 to line 4 is justified by the Subtraction Property of Equality.

Step 4: -6x = 1

To justify this step, we need to apply the Multiplication Property of Equality. This property states that if two expressions are equal, then multiplying both expressions by the same non-zero value will result in two new expressions that are also equal.

In this case, we multiply both sides of the equation -6x = 1 by -1/6. This results in:

(-1/6) * (-6x) = (-1/6) * 1

Simplifying the equation, we get:

x = 1/6

Therefore, the step from line 4 to line 5 is justified by the Multiplication Property of Equality.

Conclusion

In conclusion, the algebraic manipulation is justified by the properties of equality. Each step of the manipulation is justified by applying the appropriate property of equality.

The Final Answer

The final answer is x = 1/6.
Q&A: Algebraic Manipulation and Properties of Equality

Understanding Algebraic Manipulation

Algebraic manipulation is a crucial concept in mathematics that involves simplifying and solving equations. It requires a deep understanding of the properties of equality, which are the foundation of algebraic manipulation.

Frequently Asked Questions

Here are some frequently asked questions about algebraic manipulation and properties of equality:

Q: What is the Addition Property of Equality?

A: The Addition Property of Equality states that if two expressions are equal, then adding the same value to both expressions will result in two new expressions that are also equal.

Q: What is the Subtraction Property of Equality?

A: The Subtraction Property of Equality states that if two expressions are equal, then subtracting the same value from both expressions will result in two new expressions that are also equal.

Q: What is the Multiplication Property of Equality?

A: The Multiplication Property of Equality states that if two expressions are equal, then multiplying both expressions by the same non-zero value will result in two new expressions that are also equal.

Q: What is the Division Property of Equality?

A: The Division Property of Equality states that if two expressions are equal, then dividing both expressions by the same non-zero value will result in two new expressions that are also equal.

Q: How do I apply the properties of equality in algebraic manipulation?

A: To apply the properties of equality in algebraic manipulation, you need to identify the property that is being used in each step of the manipulation. For example, if you are subtracting the same value from both sides of an equation, you are using the Subtraction Property of Equality.

Q: What is the difference between the Addition Property of Equality and the Subtraction Property of Equality?

A: The Addition Property of Equality and the Subtraction Property of Equality are both used to simplify equations, but they are used in different ways. The Addition Property of Equality is used to add the same value to both sides of an equation, while the Subtraction Property of Equality is used to subtract the same value from both sides of an equation.

Q: How do I know which property of equality to use in a given situation?

A: To determine which property of equality to use in a given situation, you need to analyze the equation and identify the operation that is being performed. For example, if you are adding the same value to both sides of an equation, you are using the Addition Property of Equality.

Q: What is the importance of understanding the properties of equality in algebraic manipulation?

A: Understanding the properties of equality is crucial in algebraic manipulation because it allows you to simplify and solve equations. Without a deep understanding of the properties of equality, you may not be able to simplify equations correctly, which can lead to incorrect solutions.

Q: Can you provide an example of how to apply the properties of equality in algebraic manipulation?

A: Here is an example of how to apply the properties of equality in algebraic manipulation:

Suppose we have the equation:

2x + 5 = 3x - 2

To simplify this equation, we can use the Subtraction Property of Equality to subtract 2x from both sides of the equation. This results in:

5 = x - 2

Next, we can use the Addition Property of Equality to add 2 to both sides of the equation. This results in:

7 = x

Therefore, the solution to the equation is x = 7.

Q: What are some common mistakes to avoid when applying the properties of equality in algebraic manipulation?

A: Some common mistakes to avoid when applying the properties of equality in algebraic manipulation include:

  • Not identifying the property of equality that is being used in each step of the manipulation
  • Not simplifying the equation correctly
  • Not checking the solution to the equation
  • Not using the correct property of equality in a given situation

Q: How can I practice applying the properties of equality in algebraic manipulation?

A: To practice applying the properties of equality in algebraic manipulation, you can try the following:

  • Start with simple equations and gradually move on to more complex equations
  • Practice simplifying equations using the Addition Property of Equality, the Subtraction Property of Equality, the Multiplication Property of Equality, and the Division Property of Equality
  • Use online resources or algebraic manipulation software to practice solving equations
  • Work with a tutor or teacher to get feedback on your work and to learn from your mistakes

By following these tips and practicing regularly, you can become proficient in applying the properties of equality in algebraic manipulation.