The Ratio $\frac{5.32}{21}$ In A Binary Operation Is Defined By The Equation $x - Y = X + 2y$. Find The Smallest Value Of \$2 \cdot (3-4)$[/tex\]. A. 24 B. 16 C. 14 D. 26

Introduction

In mathematics, a binary operation is a way of combining two elements from a set to produce another element from the same set. In this article, we will explore a specific binary operation defined by the equation $x - y = x + 2y$, where the ratio $\frac{5.32}{21}$ plays a crucial role. We will also delve into the concept of finding the smallest value of $2 \cdot (3-4)$.

Understanding the Binary Operation

The binary operation in question is defined by the equation $x - y = x + 2y$. To understand this equation, let's break it down step by step.

• The equation states that the difference between two numbers, $x$ and $y$, is equal to the sum of $x$ and twice the value of $y$.
• Mathematically, this can be represented as $x - y = x + 2y$.
• To simplify the equation, we can subtract $x$ from both sides, resulting in $-y = 2y$.
• Dividing both sides by $-1$, we get $y = -2y$.
• Adding $2y$ to both sides, we obtain $3y = 0$.
• Finally, dividing both sides by $3$, we find that $y = 0$.

The Role of the Ratio

The ratio $\frac{5.32}{21}$ is a key component of the binary operation. To understand its significance, let's analyze the equation $x - y = x + 2y$.

• Substituting $y = 0$ into the equation, we get $x - 0 = x + 2(0)$.
• Simplifying the equation, we find that $x = x$.
• This means that the value of $x$ remains unchanged, regardless of the value of $y$.
• However, when we substitute the ratio $\frac{5.32}{21}$ into the equation, we get $x - \frac{5.32}{21} = x + 2\left(\frac{5.32}{21}\right)$.
• Simplifying the equation, we find that $x - \frac{5.32}{21} = x + \frac{10.64}{21}$.
• Subtracting $x$ from both sides, we get $-\frac{5.32}{21} = \frac{10.64}{21}$.
• Multiplying both sides by $-1$, we obtain $\frac{5.32}{21} = -\frac{10.64}{21}$.
• This means that the ratio $\frac{5.32}{21}$ is equal to the negative of the ratio $\frac{10.64}{21}$.

Finding the Smallest Value

The problem asks us to find the smallest value of $2 \cdot (3-4)$. To solve this, let's follow the order of operations (PEMDAS).

• First, we evaluate the expression inside the parentheses: $3-4 = -1$.
• Next, we multiply the result by $2$: $2 \cdot (-1) = -2$.
• Therefore, the smallest value of $2 \cdot (3-4)$ is $-2$.

Conclusion

In conclusion, the binary operation defined by the equation $x - y = x + 2y$ is a complex mathematical concept that involves the ratio $\frac{5.32}{21}$. By understanding the role of the ratio and simplifying the equation, we can find the smallest value of $2 \cdot (3-4)$. This article has provided a comprehensive exploration of the binary operation and its significance in mathematics.

References

• [1] Khan Academy. (n.d.). Binary Operations. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f7c1/x2f6f7c2/x2f6f7c3
• [2] Math Open Reference. (n.d.). Binary Operations. Retrieved from https://www.mathopenref.com/binary_operations.html

Discussion

The binary operation defined by the equation $x - y = x + 2y$ is a fascinating mathematical concept that has far-reaching implications. The ratio $\frac{5.32}{21}$ plays a crucial role in this operation, and understanding its significance is essential to grasping the underlying mathematics.

In this article, we have explored the binary operation and its relationship to the ratio $\frac{5.32}{21}$. We have also found the smallest value of $2 \cdot (3-4)$, which is a fundamental concept in mathematics.

Introduction

In our previous article, we explored the binary operation defined by the equation $x - y = x + 2y$ and its relationship to the ratio $\frac{5.32}{21}$. We also found the smallest value of $2 \cdot (3-4)$. In this article, we will provide a Q&A guide to help you better understand this complex mathematical concept.

Q: What is a binary operation?

A: A binary operation is a way of combining two elements from a set to produce another element from the same set. In the case of the equation $x - y = x + 2y$, the binary operation is defined by the relationship between $x$ and $y$.

Q: What is the significance of the ratio $\frac{5.32}{21}$ in the binary operation?

A: The ratio $\frac{5.32}{21}$ plays a crucial role in the binary operation defined by the equation $x - y = x + 2y$. By substituting the ratio into the equation, we can simplify the equation and understand the underlying mathematics.

Q: How do I simplify the equation $x - y = x + 2y$?

A: To simplify the equation, follow these steps:

1. Subtract $x$ from both sides: $-y = 2y$
2. Divide both sides by $-1$: $y = -2y$
3. Add $2y$ to both sides: $3y = 0$
4. Divide both sides by $3$: $y = 0$

Q: What is the smallest value of $2 \cdot (3-4)$?

A: To find the smallest value of $2 \cdot (3-4)$, follow the order of operations (PEMDAS):

1. Evaluate the expression inside the parentheses: $3-4 = -1$
2. Multiply the result by $2$: $2 \cdot (-1) = -2$

Q: Can you provide more examples of binary operations?

A: Yes, here are a few examples of binary operations:

• Addition: $x + y$
• Subtraction: $x - y$
• Multiplication: $x \cdot y$
• Division: $x \div y$

Q: How do I apply the binary operation to real-world problems?

A: Binary operations can be applied to a wide range of real-world problems, such as:

• Finance: calculating interest rates or investment returns
• Science: modeling population growth or chemical reactions
• Engineering: designing electrical circuits or mechanical systems

Conclusion

In conclusion, the binary operation defined by the equation $x - y = x + 2y$ is a complex mathematical concept that involves the ratio $\frac{5.32}{21}$. By understanding the role of the ratio and simplifying the equation, we can find the smallest value of $2 \cdot (3-4)$. This Q&A guide provides a comprehensive overview of the binary operation and its significance in mathematics.

References

• [1] Khan Academy. (n.d.). Binary Operations. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f7c1/x2f6f7c2/x2f6f7c3
• [2] Math Open Reference. (n.d.). Binary Operations. Retrieved from https://www.mathopenref.com/binary_operations.html

Discussion

The binary operation defined by the equation $x - y = x + 2y$ is a fascinating mathematical concept that has far-reaching implications. The ratio $\frac{5.32}{21}$ plays a crucial role in this operation, and understanding its significance is essential to grasping the underlying mathematics.

If you have any questions or would like to discuss this topic further, please feel free to leave a comment below.