Solve For X^3 - 3y: X=-3, Y=4

Dec 12, 2025 by Admin 30 views

Solve for $x^3-3y$ when $x=-3$ and $y=4$

Hey guys! Today, we're diving deep into a fun little math problem where we need to evaluate an expression given specific values for our variables. It might sound a bit technical, but trust me, it's like solving a puzzle, and once you get the hang of it, it's super satisfying. We're looking at the expression $x^3 - 3y$ , and we're given that $x$ equals -3 and $y$ equals 4. Our mission, should we choose to accept it, is to find out what this expression boils down to. We've also got some options: A. -39, B. -15, C. 15, and D. 39. So, stick around, grab your favorite thinking cap, and let's break this down step-by-step.

Understanding the Basics of Evaluating Expressions

Before we jump into the nitty-gritty of our specific problem, let's quickly chat about what it means to evaluate an expression. In mathematics, an expression is basically a combination of numbers, variables (like our $x$ and $y$ here), and operation symbols (like the minus sign and the exponent). When we're asked to evaluate an expression, it means we need to substitute the given numerical values for the variables and then perform the indicated operations to arrive at a single numerical answer. Think of it like a recipe; the expression is the set of instructions, and the variable values are the ingredients. You plug in the ingredients, follow the steps, and get your final dish. It's a fundamental skill in algebra, and mastering it opens up a whole world of more complex problem-solving. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right), is our trusty guide here. It ensures that we all get the same answer, no matter how we approach the calculation. So, in our case, $x^3 - 3y$ is the expression. We've got $x = -3$ and $y = 4$ . We'll substitute these numbers in and then follow PEMDAS to solve it. Easy peasy, right?

Plugging in the Values: Substituting $x$ and $y$

Alright, let's get down to business with our specific problem: evaluate $x^3 - 3y$ when $x = -3$ and $y = 4$ . The first step, as we discussed, is substitution. We take our expression, $x^3 - 3y$ , and wherever we see an $x$ , we're going to replace it with -3. And wherever we see a $y$ , we'll replace it with 4. It's crucial to be careful here, especially with negative numbers. When substituting a negative number, it's often a good idea to use parentheses to avoid any confusion. So, our expression becomes: $(-3)^3 - 3(4)$ . See how we put -3 in parentheses? This is super important when dealing with exponents, as it clarifies that the entire number -3 is being cubed. Likewise, putting 4 in parentheses after the 3, like $3(4)$ , also helps emphasize that we are multiplying 3 by 4. This careful substitution is the bedrock of accurate calculation. If you mess this part up, the rest of your work will be off. So, take a deep breath, double-check your substitutions, and make sure everything is exactly where it should be. We've successfully translated the abstract variables into concrete numbers. The next step is to crunch these numbers using the power of PEMDAS!

Applying the Order of Operations (PEMDAS)

Now that we've got our expression with the numbers plugged in, $(-3)^3 - 3(4)$ , it's time to apply the order of operations (PEMDAS). Remember PEMDAS? Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Let's go through it step-by-step.

First up is Parentheses. In our expression, $(-3)^3 - 3(4)$ , the parentheses are already serving their purpose by enclosing the numbers we substituted. We don't have any operations inside the parentheses that need simplifying beyond what's already there (like a sum or difference). So, we move on.

Next is Exponents. We have $(-3)^3$ . This means we need to multiply -3 by itself three times: $(-3) imes (-3) imes (-3)$ . Let's break that down: $(-3) imes (-3)$ equals a positive 9 (because a negative times a negative is a positive). Then, we take that positive 9 and multiply it by the remaining -3: $9 imes (-3)$ . A positive times a negative gives us a negative result. So, $9 imes (-3) = -27$ . Great! We've handled the exponent part. Our expression now looks like: $-27 - 3(4)$ .

Now, we move to Multiplication and Division. We have one multiplication operation here: $3(4)$ , which is the same as $3 imes 4$ . That equals 12. So, our expression becomes: $-27 - 12$ .

Finally, we have Addition and Subtraction. We are left with $-27 - 12$ . This is a subtraction problem involving two negative numbers. Think of it as starting at -27 on a number line and moving 12 units further to the left. Or, you can think of it as adding two numbers with the same sign: $-(27 + 12)$ . Either way, we get $-39$ .

So, by carefully following PEMDAS, we've calculated that $x^3 - 3y$ evaluates to -39 when $x = -3$ and $y = 4$ . Pretty straightforward when you break it down, right? We've successfully navigated the exponents and the multiplications to arrive at our final answer.

Comparing with the Options and Final Answer

We've done the math, and our calculation has led us to the answer -39. Now, let's compare our result with the given options to ensure we've got the right one. The options provided were: A. -39, B. -15, C. 15, D. 39.

Our calculated value is -39. Looking at the options, we can see that option A matches our result perfectly. This gives us confidence that our step-by-step process, including the careful substitution and application of the order of operations (PEMDAS), was accurate. It's always a good practice to quickly review your steps, especially if you have the time. Did we substitute correctly? Yes, $(-3)^3 - 3(4)$ . Did we handle the exponent correctly? Yes, $(-3)^3 = -27$ . Did we perform the multiplication correctly? Yes, $3(4) = 12$ . And finally, did we do the subtraction correctly? Yes, $-27 - 12 = -39$ . Everything checks out!

Therefore, the correct answer when evaluating $x^3 - 3y$ for $x = -3$ and $y = 4$ is -39. This corresponds to option A. It's always a great feeling to solve a problem and confirm your answer against the choices. This problem tested our understanding of substitution and the order of operations, particularly with negative numbers and exponents, which are common areas where mistakes can happen. By being methodical and double-checking each step, we were able to arrive at the correct solution. Keep practicing these types of problems, guys, and you'll become a math wizard in no time!

Conclusion: Mastering Algebraic Evaluation

In conclusion, we've successfully tackled the problem of evaluating the algebraic expression $x^3 - 3y$ by substituting the given values $x = -3$ and $y = 4$ . We meticulously followed the order of operations (PEMDAS), paying close attention to the exponentiation of a negative number and the subsequent subtraction. The process involved:

Substitution: Replacing $x$ with -3 and $y$ with 4, resulting in $(-3)^3 - 3(4)$ .
Exponentiation: Calculating $(-3)^3$ which equals -27.
Multiplication: Calculating $3(4)$ which equals 12.
Subtraction: Performing $-27 - 12$ , which yields -39.

Our final answer, -39, matched option A. This exercise highlights the importance of precision in mathematical calculations, especially when dealing with negative numbers and the rules of exponents. Understanding and applying PEMDAS correctly is fundamental to solving algebraic problems accurately. Remember, guys, math is all about building blocks. Mastering these basic evaluation skills will set you up for success in more complex algebraic concepts down the line. Keep practicing, stay curious, and don't be afraid to break down problems into smaller, manageable steps. You've got this!