Well, well, well, solving systems of equations, huh? That's like playing a little math puzzle! Let's have some fun with elimination, shall we?
First, let's put on our detective hats and investigate what we can do here. We have three equations involving three variables: x, y, and z. Our goal is to find the values that make all three equations true. To make things easier, we'll use the method of elimination to eliminate one variable at a time.
Let's start with the x's. We notice that in the first equation, there's a -2x term, and in the second equation, there's also a -2x term. What a coincidence! We can eliminate the x's just like that!
If we subtract the second equation from the first equation (because we're trying to eliminate), we get:
(-2x + 2y + 3z) - (-2x - y + z) = (0) - (-3)
Simplifying this a bit, we get:
(-2x + 2x + 2y + y + 3z - z) = 0 + 3
Oh boy, look at that! The x's cancel out, just like good old magic! Now let's simplify this equation further:
(2y + y + 3z - z) = 3
Combining like terms, we end up with:
3y + 2z = 3
Now we've gotten rid of the x's and cooked up a new equation! Time for the next act, eliminating the y's!
We notice that the third equation has a 3y term, so we can subtract the third equation from the second equation to eliminate the y variable. Here's what it looks like:
(-2x - y + z) - (2x + 3y + 3z) = -3 - 5
Oh, joy! Let's simplify this equation and see what it brings us:
(-2x - 2x - y - 3y + z - 3z) = -8
Let's clean it up a bit and combine similar terms:
(-4x - 4y - 2z) = -8
And there we have it, our second shiny new equation from eliminating the y's! Now there's only one variable left standing, and that is z.
In the first equation, we can see that there's a 3z term, and in the second equation, there's a -3z term. Lovely! It's time for their ultimate showdown!
By subtracting the first equation from the third equation, we get:
(2x + 3y + 3z) - (-2x + 2y + 3z) = 5 - 0
Let's simplify this a tad:
(2x + 2x + 3y - 2y + 3z - 3z) = 5
After combining like terms, we have:
4x + y = 5
Oh, snap! We've thrown out the z's and discovered our third and final equation.
Now it's time to put on our detective goggles again and analyze the situation. We now have three fresh equations:
3y + 2z = 3 (from eliminating x)
-4x - 4y - 2z = -8 (from eliminating y)
4x + y = 5 (from eliminating z)
Now, it's time for the grand finale! We can solve this system of equations using various methods, such as substitution or matrix methods.
But, hey, since we're all about the fun and humor, let's leave the final answer up to you! Solve these equations, and may the math gods be ever on your side, my friend! Good luck!