Evaluating (2x^2 - 5y) / (3x - Y) For X=2 And Y=-4
In mathematics, evaluating expressions is a fundamental skill. It involves substituting given values for variables and simplifying the resulting arithmetic expression. This article focuses on the evaluation of the algebraic expression (2x^2 - 5y) / (3x - y) when x = 2 and y = -4. This type of problem is commonly encountered in algebra and is crucial for building a strong foundation in mathematical problem-solving. We will break down the steps involved in the evaluation, providing a clear and concise explanation to ensure understanding. This will involve substituting the given values, performing the necessary arithmetic operations (including exponents, multiplication, and subtraction), and finally, simplifying the expression to arrive at the final answer. By understanding this process, you'll be better equipped to tackle more complex algebraic expressions and problems.
Understanding the Expression
Before we begin the evaluation, it's essential to understand the expression itself. The expression (2x^2 - 5y) / (3x - y) is a rational expression, meaning it is a fraction where the numerator and denominator are polynomials. The numerator is 2x^2 - 5y, which involves a term with x squared and a term with y. The denominator is 3x - y, a simpler linear expression. To evaluate this expression for specific values of x and y, we will substitute the given values into the expression and simplify using the order of operations (PEMDAS/BODMAS).
The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), is a set of rules that dictate the sequence in which mathematical operations should be performed. This ensures that mathematical expressions are evaluated consistently and unambiguously. In the expression (2x^2 - 5y) / (3x - y), we will first deal with the exponent in the term 2x^2, then perform the multiplications in both the numerator and the denominator, and finally, carry out the subtractions. The entire numerator and denominator will be simplified separately before the final division is performed. Adhering to this order is critical for obtaining the correct result. Without a clear understanding of the order of operations, the evaluation of mathematical expressions can easily lead to errors. This principle is not only fundamental in algebra but is also crucial in more advanced mathematical fields and real-world applications.
Step-by-Step Evaluation
Now, let's proceed with the step-by-step evaluation of the expression (2x^2 - 5y) / (3x - y) when x = 2 and y = -4. This process will involve substitution, simplification, and careful attention to the order of operations.
1. Substitution
The first step is to substitute the given values of x and y into the expression. Replace every instance of x with 2 and every instance of y with -4. This yields the following:
(2 * (2)^2 - 5 * (-4)) / (3 * 2 - (-4))
This substitution step is critical because it transforms the algebraic expression into a purely arithmetic one, which we can then simplify using the order of operations. Ensuring that the values are substituted correctly is paramount, as any error in this step will propagate through the rest of the calculation. This initial substitution sets the stage for the subsequent steps and is the foundation upon which the solution is built.
2. Simplifying the Numerator
The numerator of the expression is 2 * (2)^2 - 5 * (-4). Following the order of operations (PEMDAS/BODMAS), we first address the exponent:
(2)^2 = 4
Now the numerator becomes:
2 * 4 - 5 * (-4)
Next, perform the multiplications:
2 * 4 = 8
5 * (-4) = -20
So the numerator is now:
8 - (-20)
Finally, perform the subtraction. Remember that subtracting a negative number is the same as adding its positive counterpart:
8 - (-20) = 8 + 20 = 28
Therefore, the simplified numerator is 28. Each of these steps adheres strictly to the order of operations, ensuring the accurate simplification of the numerator. Paying close attention to the signs and the order in which the operations are performed is key to avoiding errors. The simplification of the numerator is a crucial component of evaluating the entire expression, and mastering this process is essential for algebraic manipulation.
3. Simplifying the Denominator
The denominator of the expression is 3 * 2 - (-4). Following the order of operations, we first perform the multiplication:
3 * 2 = 6
Now the denominator becomes:
6 - (-4)
Next, perform the subtraction. Again, subtracting a negative number is the same as adding its positive counterpart:
6 - (-4) = 6 + 4 = 10
Therefore, the simplified denominator is 10. Just as with the numerator, the simplification of the denominator requires careful adherence to the order of operations and attention to the signs. This step is equally important in determining the final result of the expression. The ability to accurately simplify denominators is a fundamental skill in algebra and is essential for working with rational expressions.
4. Final Calculation
Now that we have simplified both the numerator and the denominator, we can perform the final division. The expression is now:
28 / 10
This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
(28 / 2) / (10 / 2) = 14 / 5
Therefore, the final value of the expression (2x^2 - 5y) / (3x - y) when x = 2 and y = -4 is 14/5. This final step brings together the results of the previous simplifications and presents the answer in its simplest form. The ability to reduce fractions to their simplest form is an important skill in mathematics and ensures that the answer is expressed in the most concise manner.
Conclusion
In conclusion, by substituting x = 2 and y = -4 into the expression (2x^2 - 5y) / (3x - y) and following the order of operations, we have found the value to be 14/5. This process demonstrates the importance of careful substitution, adherence to the order of operations, and simplification techniques in evaluating algebraic expressions. The step-by-step breakdown provided here offers a clear and concise guide to solving similar problems. Evaluating algebraic expressions is a fundamental skill in mathematics, and mastering this process is crucial for success in algebra and beyond. Through practice and a thorough understanding of the underlying principles, you can confidently tackle more complex expressions and problems.
Practice Problems
To further solidify your understanding, try evaluating the following expressions using the same methods:
- (3a^2 + 2b) / (a - b) when a = 3 and b = -2
- (x^3 - 4y) / (2x + y^2) when x = -1 and y = 3
Working through these practice problems will help you reinforce the concepts and techniques discussed in this article. Remember to pay close attention to the order of operations and the signs of the numbers involved. Consistent practice is key to developing proficiency in evaluating algebraic expressions. By tackling a variety of problems, you will build confidence and improve your problem-solving skills.
