Evaluate Expression: X^2 + 5y - 2xy^2, X=-3, Y=5

Oct 19, 2025 by ADMIN 49 views

Evaluate the Expression: x^2 + 5y - 2xy^2 where x = -3 and y = 5

Hey guys! Let's dive into evaluating algebraic expressions. In this article, we're going to break down how to solve the expression x^2 + 5y - 2xy^2 when x = -3 and y = 5. This is a fundamental concept in algebra, and mastering it will help you tackle more complex problems down the road. So, grab your calculators (or your mental math muscles) and let's get started!

Understanding Algebraic Expressions

Before we jump into the specific problem, let's make sure we're all on the same page about what an algebraic expression is. An algebraic expression is a combination of variables (like x and y), constants (like 5 and -2), and mathematical operations (like addition, subtraction, multiplication, and exponentiation). The key here is that there's no equals sign (=), which distinguishes it from an equation. Think of it as a phrase rather than a complete sentence.

When we evaluate an algebraic expression, we're essentially finding its numerical value for specific values of the variables. This involves substituting the given values for the variables and then simplifying the expression using the order of operations (PEMDAS/BODMAS). This process is super important in various fields, from engineering to economics, where we often need to calculate results based on changing conditions. Understanding how to correctly substitute values and simplify expressions ensures accurate calculations and reliable outcomes, which is crucial in these practical applications. So, by mastering these skills, you are not only getting better at math but also preparing for real-world problem-solving scenarios.

Step-by-Step Evaluation

Okay, let's get into the nitty-gritty of our problem: x^2 + 5y - 2xy^2, where x = -3 and y = 5. We'll take this one step at a time to make sure we don't miss anything.

1. Substitution

The first step is to substitute the given values of x and y into the expression. This means replacing every instance of x with -3 and every instance of y with 5. So, our expression becomes:

(-3)^2 + 5(5) - 2(-3)(5)^2

Substitution is a crucial first step because it transforms the abstract algebraic expression into a concrete numerical expression. Think of it as translating a mathematical sentence from a language of symbols into a language of numbers. It sets the stage for the rest of the calculation by giving us specific values to work with. This step requires careful attention to detail to ensure that the correct values are placed in the correct positions, and using parentheses when substituting negative numbers is crucial to avoid sign errors. This seemingly simple act of substitution is really the foundation upon which the rest of the solution is built, and getting it right is essential for arriving at the correct final answer.

2. Exponents

Next up, we need to deal with any exponents. In our expression, we have (-3)^2 and (5)^2. Remember that squaring a number means multiplying it by itself.

(-3)^2 = (-3) * (-3) = 9
(5)^2 = 5 * 5 = 25

So, our expression now looks like this:

9 + 5(5) - 2(-3)(25)

Handling exponents early is crucial because they represent repeated multiplication, and these operations have a higher priority than multiplication, division, addition, and subtraction. By addressing exponents first, we ensure that the order of operations is followed correctly, preventing errors that can arise from performing other operations prematurely. For example, if we were to multiply before squaring, we would completely change the outcome of the calculation. This step not only simplifies the expression but also sets the stage for the remaining operations, leading us closer to the final solution in a structured and mathematically sound manner. Getting the exponents right is a cornerstone of accurate algebraic manipulation.

3. Multiplication

Now we move on to multiplication. We have two multiplications to take care of: 5(5) and 2(-3)(25). Let's do them one at a time:

5(5) = 25
2(-3)(25) = -6(25) = -150

Our expression is now:

9 + 25 - (-150)

Performing multiplication before addition and subtraction is a key aspect of the order of operations (PEMDAS/BODMAS). Multiplication represents a more complex operation than addition or subtraction, and handling it at this stage ensures that we correctly scale the values before combining them. This step not only simplifies the expression further but also sets the stage for the final addition and subtraction steps. By dealing with multiplication in its proper sequence, we maintain the mathematical integrity of the expression, ensuring that each operation is performed in the correct context and contributes accurately to the final result.

4. Addition and Subtraction

Finally, we're left with addition and subtraction. Remember that subtracting a negative number is the same as adding its positive counterpart. So, we have:

9 + 25 - (-150) = 9 + 25 + 150

Now, let's add the numbers together:

9 + 25 + 150 = 34 + 150 = 184

So, the value of the expression x^2 + 5y - 2xy^2 when x = -3 and y = 5 is 184.

Performing addition and subtraction last is crucial in the order of operations because these operations combine the results of the earlier steps. Addition and subtraction finalize the calculation by bringing all the terms together into a single value. By saving these operations for the end, we ensure that all multiplications, divisions, and exponentiations have been properly accounted for, leading to a correct and consolidated final answer. This sequencing of operations is fundamental to mathematical accuracy and is vital for problem-solving in a wide range of contexts.

Common Mistakes to Avoid

When evaluating expressions, there are a few common pitfalls that students often fall into. Let's highlight some of these so you can steer clear of them:

Forgetting the order of operations: Always remember PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). Doing operations out of order will lead to incorrect results.
Sign errors: Be extra careful when dealing with negative numbers, especially when squaring them or multiplying them. A small mistake with a sign can throw off the entire calculation.
Incorrect substitution: Double-check that you've substituted the values correctly. It's easy to mix up the values of x and y or to forget a negative sign.
Skipping steps: It's tempting to rush through the calculation, but taking it one step at a time minimizes the chance of errors. Write out each step clearly.

By recognizing these common mistakes, you can develop strategies to avoid them. For instance, writing out each step explicitly helps to keep track of your work and reduces the likelihood of overlooking an operation or making a sign error. Double-checking your substitutions ensures that you’re working with the correct values from the start. Understanding and diligently applying the order of operations becomes second nature with practice, preventing missteps in the sequence of calculations. Paying attention to these details can significantly improve your accuracy and confidence in evaluating algebraic expressions.

Practice Makes Perfect

The best way to master evaluating expressions is to practice, practice, practice! Try working through similar problems with different values for the variables. You can also create your own expressions and challenge yourself or your friends to solve them. The more you practice, the more comfortable and confident you'll become.

To further improve your skills, you might look for online resources or textbooks that offer a variety of practice problems and step-by-step solutions. Working through a wide range of examples helps you to see different applications of the same principles, solidifying your understanding. Collaborative problem-solving, where you work with peers or tutors, can also be incredibly beneficial. Explaining your thought process and working through difficulties together can highlight areas where you might be making mistakes and introduce you to alternative problem-solving strategies. Remember, each problem you solve is a step towards mastery, and the effort you put in now will pay off in your future mathematical endeavors.

Conclusion

So, there you have it! We've successfully evaluated the expression x^2 + 5y - 2xy^2 for x = -3 and y = 5. Remember to take it one step at a time, follow the order of operations, and watch out for those common mistakes. With a little practice, you'll be evaluating expressions like a pro in no time!

Evaluating algebraic expressions is a fundamental skill in mathematics that opens doors to more complex topics and real-world applications. By mastering this skill, you're not just learning a procedure; you're developing a way of thinking that is crucial for problem-solving in many fields. The ability to break down a complex expression into manageable parts, to substitute values accurately, and to follow a logical sequence of operations is a powerful tool. This proficiency builds confidence in your mathematical abilities and prepares you to tackle increasingly challenging problems. So, keep practicing, keep exploring, and embrace the journey of learning mathematics – it’s a journey that will take you far!