Solving Equations: Find (x+z)^y With Real Numbers

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Hey guys! Let's dive into a cool math problem. We're given a system of equations with real numbers x, y, and z. Our goal? To figure out the value of (x+z)^y. Sounds like fun, right?

Setting Up the Problem: Understanding the Equations

Alright, so here's what we've got. We have three equations:

  1. x - 2y - 2z = -1
  2. x - 2z = 3
  3. y - z = 1

Our task is to find the value of the expression (x+z)^y. To do this, we need to first solve for x, y, and z. These kinds of problems are super common in algebra, and the key is to be systematic. Don't worry, it's not as scary as it looks! We'll go through it step by step, and I'll break it down so it's easy to follow.

The first thing we want to do when tackling a system of equations is to try to simplify things. Look for the simplest equations first. In our case, equations 2 and 3 look like good starting points. Equation 2 gives us a direct relationship between x and z, and equation 3 tells us the relationship between y and z. This is great because it gives us a way to find the variables' values one by one. We'll use substitution to solve this, which means we'll solve for one variable in terms of another and then plug that into another equation. Stick with me, and you'll become a pro at these in no time!

Let's start with equation 3: y - z = 1. We can easily solve for y in terms of z: y = z + 1. This is our first key piece of the puzzle. Now, let's turn our attention to equation 2: x - 2z = 3. This one is also straightforward, and we can solve for x in terms of z: x = 2z + 3. Fantastic! We have x and y expressed in terms of z. This means we can substitute these values into equation 1 to solve for z.

Solving for x, y, and z: The Substitution Method

Okay, now we're going to use the substitution method, which is one of the best ways to solve systems of equations, especially when they're set up like this. We've already done a bit of the groundwork by expressing x and y in terms of z. Now, let's plug these expressions into equation 1, which is: x - 2y - 2z = -1. Remember, we have:

  • x = 2z + 3
  • y = z + 1

Substituting these into equation 1, we get: (2z + 3) - 2(z + 1) - 2z = -1. See how we've replaced x and y with their equivalents in terms of z? Now, it's just a matter of simplifying and solving for z.

Let's simplify the equation: 2z + 3 - 2z - 2 - 2z = -1. Combining like terms, we have: -2z + 1 = -1. Now, subtract 1 from both sides: -2z = -2. Finally, divide both sides by -2: z = 1. Yay! We've found the value of z! Now that we have z, we can easily find x and y.

We know that x = 2z + 3. Since z = 1, then x = 2(1) + 3 = 2 + 3 = 5. So, x = 5. And from y = z + 1, with z = 1, we get y = 1 + 1 = 2. Therefore, y = 2. So, we've solved for all three variables: x = 5, y = 2, z = 1.

Calculating (x+z)^y: Putting It All Together

Alright, we're on the home stretch! We've successfully found the values of x, y, and z. Now, all we need to do is plug these values into the expression (x+z)^y and calculate the result. Remember, we have: x = 5, y = 2, z = 1. The expression we need to solve is (x+z)^y. Substituting the values of x, y, and z, we have: (5 + 1)^2.

First, simplify inside the parentheses: 5 + 1 = 6. So, the expression becomes: 6^2. And what is 6 squared? 6 * 6 = 36. And that's our answer! So, (x+z)^y = 36. We did it, guys! We successfully solved a system of equations and found the value of the expression. This is a great example of how understanding algebra and being systematic can help us solve complex problems. Keep practicing, and you'll become a master of this stuff in no time. Remember to break down the problems into smaller steps, solve for one variable at a time, and always double-check your work!

Conclusion: Mastering Equation Solving

So, there you have it! We've successfully solved for x, y, and z and found that (x+z)^y equals 36. This problem illustrates the power of substitution and simplification in solving systems of equations. The key takeaways here are:

  • Identify the Equations: Recognize the given equations and the expression you need to evaluate.
  • Simplify and Substitute: Use substitution to express variables in terms of each other and reduce the number of variables in each equation.
  • Solve Step-by-Step: Take it one step at a time, solving for one variable at a time.
  • Check Your Work: Always double-check your calculations to avoid silly mistakes.

Understanding and applying these concepts will help you tackle many other math problems. Keep practicing, and you'll get better and faster at solving these kinds of problems. Remember, math is all about practice and consistency! The more you work through problems, the more confident you'll become. So, keep up the great work, and keep exploring the fascinating world of mathematics. Also, feel free to reach out if you have any more questions. Keep solving, keep learning, and keep the mathematical journey going, my friends! I hope this helps. Let me know if you have any other questions. Good luck!