Solving A System Of Equations And Evaluating Statements

Let's dive into solving this system of equations and checking out those statements. It's like a mini-math adventure, so grab your thinking caps, guys!

Solving the System of Equations

We've got three equations:

1. x + y + z = 6
2. x - y + z = 2
3. x + y - z = 0

First, let's eliminate 'y' by adding equations (1) and (2):

(x + y + z) + (x - y + z) = 6 + 2 2x + 2z = 8 x + z = 4 (Divide by 2)

Let's call this equation (4): x + z = 4

Next, let's eliminate 'y' again, this time by adding equations (2) and (3):

(x - y + z) + (x + y - z) = 2 + 0 2x = 2 x = 1

Now that we know x = 1, we can plug it into equation (4):

1 + z = 4 z = 3

Finally, let's substitute x = 1 and z = 3 into equation (1) to find 'y':

1 + y + 3 = 6 y + 4 = 6 y = 2

So, the solution to the system of equations is (x, y, z) = (1, 2, 3). This is the foundation for evaluating the truthfulness of the statements, it is important to calculate correctly to ensure the correctness of the next steps. The correct calculation will greatly assist in solving the problem and lead to the right answer. This stage requires carefulness and accuracy to avoid mistakes that can affect the final result. With the solution clearly determined, we can now confidently move on to evaluating the given statements. Understanding the solution thoroughly will ensure that the evaluation is accurate and reliable. This approach ensures that each step is carefully considered, leading to a more precise and trustworthy conclusion. This careful process also allows for better understanding and easier explanation of the results. So, with the solution in hand, we can proceed to analyze the truthfulness of each statement. This strategic approach provides a solid base for further analysis and ensures that each conclusion is well-supported. The determination of (x, y, z) = (1, 2, 3) is crucial for the accurate assessment of the statements, so let's keep it in mind as we proceed.

Evaluating the Statements

Now, let's check if the following statements are true or false, given that (x, y, z) = (1, 2, 3).

Statement 1: y = x + 1

Substitute the values of x and y:

2 = 1 + 1 2 = 2

This statement is TRUE. Yes, it checks out! It's always satisfying when the math works perfectly.

Statement 2: z = x + y

Substitute the values of x, y, and z:

3 = 1 + 2 3 = 3

This statement is also TRUE. Another win! It seems like our solution is pretty solid.

Statement 3: x > y

Substitute the values of x and y:

1 > 2

This statement is FALSE. Nope, 1 is definitely not greater than 2. This one didn't hold up, but that's okay! It's all part of the process.

To summarize, we've meticulously evaluated each statement against the solution we derived for the system of equations. Statement 1, y = x + 1, holds true as 2 equals 1 + 1. Similarly, Statement 2, z = x + y, is also true because 3 is indeed the sum of 1 and 2. However, Statement 3, x > y, is false since 1 is not greater than 2. This comprehensive evaluation provides a clear understanding of the relationship between the variables in the context of the given statements. By substituting the values of x, y, and z into each statement, we were able to definitively determine their truthfulness. This methodical approach ensures accuracy and leaves no room for ambiguity in our conclusions. Understanding the nuances of each statement and how they relate to the solution is key to grasping the overall concept. This exercise highlights the importance of careful substitution and logical reasoning in mathematical problem-solving. The combination of solving equations and evaluating statements offers a well-rounded understanding of the principles involved. The process not only reinforces our understanding of algebraic manipulation but also enhances our critical thinking skills. This analytical approach is invaluable in tackling more complex problems and gaining a deeper appreciation for mathematical concepts. So, let's continue to apply these techniques and strive for accuracy and clarity in our mathematical endeavors.

Final Answers

Here's the breakdown in the table format:

And there you have it! We've successfully solved the system of equations and figured out which statements are true and which are false. Solving systems of equations and then using those solutions to evaluate statements is a really useful skill in math. It helps you understand how different parts of an equation relate to each other and how to use that knowledge to make logical deductions. The process also teaches you to be precise and detail-oriented, which are important qualities in any field. Remember, math isn't just about numbers and symbols; it's about problem-solving, critical thinking, and making connections. These are the skills that will help you succeed not only in your math classes but also in your everyday life. So, keep practicing, keep exploring, and keep asking questions. The more you engage with math, the more comfortable and confident you'll become. And who knows, maybe one day you'll be solving even more complex problems and sharing your knowledge with others. The possibilities are endless, so embrace the challenge and enjoy the journey!