Evaluating 36x-8y^2 When X Is 3 And Y Is -6

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Hey guys! Today, we're diving into a fun little math problem. We're going to figure out the value of the expression 36x - 8y² when x is 3 and y is -6. Sounds like a piece of cake, right? Let's break it down step by step so everyone can follow along. This is a classic algebra problem that combines substitution and the order of operations, so it's a great way to flex those math muscles. Whether you're a student brushing up on your algebra skills or just someone who enjoys a good mathematical puzzle, this walkthrough will help you understand the process and arrive at the correct answer. So, grab your pencils and let's get started!

Understanding the Expression

Before we jump into plugging in the numbers, let's make sure we understand what the expression 36x - 8y² actually means. This expression is made up of a few different parts: coefficients, variables, and an exponent. The coefficients are the numbers that are multiplied by the variables. In this case, we have 36 and -8. The variables are the letters, which represent unknown values. Here, we have x and y. And finally, we have an exponent, which is the small number written above and to the right of a variable. In this expression, we have y², which means y is raised to the power of 2. Understanding these components is crucial because it dictates how we approach solving the problem. We need to remember the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), to ensure we solve the expression correctly. By recognizing the different parts of the expression, we set ourselves up for success in substituting the given values and simplifying the result. So, let's move on to the next step: plugging in those values!

Step 1: Substitute the Values

The first thing we need to do is substitute the given values of x and y into our expression. We know that x = 3 and y = -6. So, wherever we see an x in the expression, we're going to replace it with 3, and wherever we see a y, we're going to replace it with -6. This gives us:

36(3) - 8(-6)²

Notice how we've put the numbers in parentheses. This is super important, especially when dealing with negative numbers or exponents. The parentheses help us keep track of what we're multiplying and raising to powers. Now that we've substituted the values, our expression looks a little more concrete. We've gone from an abstract algebraic expression to a numerical one, which is much easier to handle. But we're not done yet! We still need to simplify this expression to get our final answer. This involves following the correct order of operations, which we'll tackle in the next step. So, let's keep the momentum going and move on to simplifying our expression!

Step 2: Apply the Order of Operations (PEMDAS/BODMAS)

Alright, now comes the fun part: applying the order of operations! You might have heard of the acronyms PEMDAS or BODMAS, which stand for:

  • Parentheses / Brackets
  • Exponents / Orders
  • Multiplication and Division
  • Addition and Subtraction

This tells us the order in which we need to perform the operations in our expression. Looking at our expression 36(3) - 8(-6)², we see that we have parentheses, an exponent, multiplication, and subtraction. According to PEMDAS/BODMAS, we need to tackle the exponent first. So, let's calculate (-6)². Remember, this means -6 multiplied by itself:

(-6)² = (-6) * (-6) = 36

Now, our expression looks like this:

36(3) - 8(36)

Next up, we have multiplication. We have two multiplication operations to perform: 36(3) and 8(36). Let's do them one at a time:

36(3) = 108

8(36) = 288

Now our expression is even simpler:

108 - 288

Finally, we have subtraction. So, let's subtract:

108 - 288 = -180

And there you have it! We've successfully simplified the expression using the order of operations. It's like solving a puzzle, one step at a time. Now, let's wrap up with our final answer and a quick recap.

Step 3: State the Final Answer

After carefully substituting the values and applying the order of operations, we've arrived at our final answer. The value of the expression 36x - 8y² when x = 3 and y = -6 is:

-180

Woohoo! We did it! It might seem like a lot of steps, but each one is crucial to getting the correct answer. It's like building a house – you need a solid foundation and each brick carefully placed to create a sturdy structure. In math, each step is like that brick, contributing to the overall solution. Now, let's do a quick recap of the entire process to make sure we've got it all down pat.

Recap

Okay, let's quickly recap what we've done to solve this problem. Remember, we started with the expression 36x - 8y² and were asked to find its value when x = 3 and y = -6. Here's a step-by-step rundown:

  1. Substitute the values: We replaced x with 3 and y with -6, giving us 36(3) - 8(-6)².
  2. Apply the order of operations (PEMDAS/BODMAS):
    • First, we tackled the exponent: (-6)² = 36.
    • Then, we performed the multiplications: 36(3) = 108 and 8(36) = 288.
    • Finally, we did the subtraction: 108 - 288 = -180.
  3. State the final answer: The value of the expression is -180.

See? When you break it down into smaller steps, it becomes much more manageable. And the best part is, you can apply this same approach to all sorts of algebraic expressions. Just remember to substitute carefully and follow the order of operations, and you'll be solving problems like a pro in no time! Now, let's talk a bit about why these kinds of problems are important and where you might encounter them in real life.

Why This Matters: Real-World Applications

Now, you might be thinking,