Given That $x = -3$ And $y = -7$, Evaluate $\frac{x^2 - Y}{y^2 - X}$.

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Introduction

In mathematics, evaluating expressions with given values is a fundamental concept that helps us solve problems and make predictions. Given that $x = -3$ and $y = -7$, we are tasked with evaluating the expression x2−yy2−x\frac{x^2 - y}{y^2 - x}. In this article, we will break down the expression, substitute the given values, and simplify the result.

Understanding the Expression

The given expression is x2−yy2−x\frac{x^2 - y}{y^2 - x}. To evaluate this expression, we need to follow the order of operations (PEMDAS):

  1. Evaluate the exponents (x2x^2 and y2y^2)
  2. Subtract yy from x2x^2
  3. Subtract xx from y2y^2
  4. Divide the result of step 2 by the result of step 3

Substituting the Given Values

We are given that $x = -3$ and $y = -7$. Let's substitute these values into the expression:

x2−yy2−x=(−3)2−(−7)(−7)2−(−3)\frac{x^2 - y}{y^2 - x} = \frac{(-3)^2 - (-7)}{(-7)^2 - (-3)}

Evaluating the Exponents

Now, let's evaluate the exponents:

(−3)2=9(-3)^2 = 9

(−7)2=49(-7)^2 = 49

Simplifying the Expression

Now that we have evaluated the exponents, let's simplify the expression:

9−(−7)49−(−3)=9+749+3=1652\frac{9 - (-7)}{49 - (-3)} = \frac{9 + 7}{49 + 3} = \frac{16}{52}

Reducing the Fraction

The fraction 1652\frac{16}{52} can be reduced by dividing both the numerator and the denominator by their greatest common divisor, which is 4:

1652=413\frac{16}{52} = \frac{4}{13}

Conclusion

In this article, we evaluated the expression x2−yy2−x\frac{x^2 - y}{y^2 - x} given that $x = -3$ and $y = -7$. We followed the order of operations, substituted the given values, evaluated the exponents, simplified the expression, and reduced the fraction. The final result is 413\frac{4}{13}.

Frequently Asked Questions

  • What is the order of operations?
    • The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS is commonly used to remember the order of operations:
      1. Parentheses
      2. Exponents
      3. Multiplication and Division
      4. Addition and Subtraction
  • How do I evaluate expressions with given values?
    • To evaluate expressions with given values, follow these steps:
      1. Substitute the given values into the expression
      2. Evaluate the exponents
      3. Simplify the expression
      4. Reduce the fraction (if necessary)

Final Thoughts

Evaluating expressions with given values is a crucial skill in mathematics. By following the order of operations and substituting the given values, we can simplify complex expressions and arrive at a final result. In this article, we evaluated the expression x2−yy2−x\frac{x^2 - y}{y^2 - x} given that $x = -3$ and $y = -7$. The final result is 413\frac{4}{13}.

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Introduction

In our previous article, we evaluated the expression x2−yy2−x\frac{x^2 - y}{y^2 - x} given that $x = -3$ and $y = -7$. We followed the order of operations, substituted the given values, evaluated the exponents, simplified the expression, and reduced the fraction. The final result was 413\frac{4}{13}. In this article, we will answer some frequently asked questions related to evaluating expressions with given values.

Q&A

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS is commonly used to remember the order of operations:

  1. Parentheses
  2. Exponents
  3. Multiplication and Division
  4. Addition and Subtraction

Q: How do I evaluate expressions with given values?

A: To evaluate expressions with given values, follow these steps:

  1. Substitute the given values into the expression
  2. Evaluate the exponents
  3. Simplify the expression
  4. Reduce the fraction (if necessary)

Q: What is the difference between evaluating an expression and simplifying an expression?

A: Evaluating an expression involves substituting given values and performing the necessary operations to arrive at a final result. Simplifying an expression involves rewriting the expression in a simpler form, often by combining like terms or canceling out common factors.

Q: How do I simplify an expression?

A: To simplify an expression, follow these steps:

  1. Combine like terms
  2. Cancel out common factors
  3. Rewrite the expression in a simpler form

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. For example, the GCD of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 without leaving a remainder.

Q: How do I find the GCD of two numbers?

A: To find the GCD of two numbers, you can use the following methods:

  1. List the factors of each number
  2. Identify the common factors
  3. Choose the largest common factor

Alternatively, you can use the Euclidean algorithm to find the GCD.

Q: What is the Euclidean algorithm?

A: The Euclidean algorithm is a method for finding the greatest common divisor (GCD) of two numbers. It involves repeatedly dividing the larger number by the smaller number and taking the remainder until the remainder is zero. The last non-zero remainder is the GCD.

Q: How do I use the Euclidean algorithm to find the GCD?

A: To use the Euclidean algorithm to find the GCD, follow these steps:

  1. Divide the larger number by the smaller number
  2. Take the remainder
  3. Replace the larger number with the smaller number and the smaller number with the remainder
  4. Repeat steps 1-3 until the remainder is zero
  5. The last non-zero remainder is the GCD

Conclusion

Evaluating expressions with given values is a crucial skill in mathematics. By following the order of operations and substituting the given values, we can simplify complex expressions and arrive at a final result. In this article, we answered some frequently asked questions related to evaluating expressions with given values. We hope this article has been helpful in clarifying any doubts you may have had.

Final Thoughts

Evaluating expressions with given values is a fundamental concept in mathematics. By mastering this skill, you can solve a wide range of problems and make predictions. Remember to follow the order of operations, substitute the given values, evaluate the exponents, simplify the expression, and reduce the fraction (if necessary). With practice and patience, you will become proficient in evaluating expressions with given values.

Additional Resources

  • Khan Academy: Evaluating Expressions
  • Mathway: Evaluating Expressions
  • Wolfram Alpha: Evaluating Expressions

Frequently Asked Questions

  • What is the order of operations?
  • How do I evaluate expressions with given values?
  • What is the difference between evaluating an expression and simplifying an expression?
  • How do I simplify an expression?
  • What is the greatest common divisor (GCD)?
  • How do I find the GCD of two numbers?
  • What is the Euclidean algorithm?
  • How do I use the Euclidean algorithm to find the GCD?