Hey everyone! Today, we're diving into a fun math problem involving a system of equations. We'll explore how to solve it and then zoom in on a specific scenario when t=0. Plus, we'll check out what happens when we add the condition x-y=0. Let's get started!
Understanding the System of Equations
So, we've got this system of equations staring back at us:
2x + 3y + 7z = 14t + 7
5x - 3y + 4z + 6y = 14t - 14
Systems of equations like these pop up all the time in various fields – think engineering, economics, and even computer science. Basically, they help us model situations where multiple variables are intertwined, and we need to figure out their values simultaneously. Our mission here is to find the values of x, y, and z that satisfy both equations at the same time. We will also be looking into the case of t=0 and how the extra condition of x-y=0 might affect our solutions. To kick things off, let's simplify these equations a bit to make them easier to work with. The first equation, 2x + 3y + 7z = 14t + 7, looks pretty straightforward. But the second one, 5x - 3y + 4z + 6y = 14t - 14, has a little something we can tidy up. Notice those 'y' terms? We've got a '-3y' and a '+6y'. Let's combine those to get a cleaner equation. When we do that, we end up with 5x + 3y + 4z = 14t - 14. Now our system looks a bit more manageable:
2x + 3y + 7z = 14t + 7
5x + 3y + 4z = 14t - 14
Simplifying Equations and Preparing for Solutions
Okay, simplifying equations is like the bread and butter of solving any math problem, especially when we're talking about systems of equations. By making things neater, we reduce the chances of making silly mistakes and make the whole solving process smoother. Now that our system looks a bit cleaner, we can start thinking about how to actually solve for x, y, and z. There are a few different ways we can tackle this. One popular method is elimination. The idea behind elimination is to strategically add or subtract the equations in a way that cancels out one of the variables. This leaves us with a system that has fewer variables, making it easier to solve. Another method is substitution. With substitution, we solve one equation for one variable in terms of the others and then substitute that expression into the other equation(s). This also reduces the number of variables we need to deal with at once. And of course, we can't forget about matrices! If you're comfortable with linear algebra, setting up the system as a matrix and using methods like Gaussian elimination can be super efficient. The best method really depends on the specific system you're dealing with, and sometimes a combination of methods is the way to go. In our case, looking at the equations, we can see that the '3y' term appears in both equations. This makes elimination a pretty attractive option. If we subtract one equation from the other, the '3y' terms will cancel out, leaving us with an equation involving just 'x' and 'z'. This is a great first step towards unraveling the system and finding our solutions.
Solving the System Using Elimination
Let's dive into solving the system using elimination. Remember our simplified system:
2x + 3y + 7z = 14t + 7
5x + 3y + 4z = 14t - 14
As we discussed, the '3y' terms are just begging to be eliminated. To do this, we'll subtract the first equation from the second equation. This means we'll take (5x + 3y + 4z) and subtract (2x + 3y + 7z) from it. On the right side, we'll do the same thing: subtract (14t + 7) from (14t - 14). Let's break it down:
(5x + 3y + 4z) - (2x + 3y + 7z) = (14t - 14) - (14t + 7)
Now, let's simplify both sides. On the left, we have 5x - 2x which gives us 3x. The 3y terms cancel out, which is exactly what we wanted! Then we have 4z - 7z, which gives us -3z. So the left side simplifies to 3x - 3z. On the right side, the 14t terms cancel out. We're left with -14 - 7, which is -21. Putting it all together, we get:
3x - 3z = -21
Hey, this looks much simpler! We can even divide both sides by 3 to make it even cleaner:
x - z = -7
This equation tells us a relationship between x and z. It says that x is always 7 less than z. This is a great piece of the puzzle, but we still need to figure out the actual values of x, y, and z. To do that, we'll need to use this new equation along with one of our original equations to eliminate another variable. We can pick either of the original equations, but let's go with the first one: 2x + 3y + 7z = 14t + 7. Now we have two equations:
x - z = -7
2x + 3y + 7z = 14t + 7
We can use these to eliminate either x or z and get an equation involving y. From there, we'll be in a good spot to find all the solutions.
Focusing on the Case When t=0
Alright, let's shift our focus to the case when t=0. This means we're looking at a specific scenario where the parameter 't' is equal to zero. Plugging t=0 into our original system of equations gives us a new, simplified system to work with. This is often a useful technique in math and physics – examining specific cases can give us valuable insights into the general behavior of the system. So, let's take our original equations:
2x + 3y + 7z = 14t + 7
5x + 3y + 4z = 14t - 14
And substitute t=0 into them. For the first equation, 14t + 7 becomes 14(0) + 7, which is simply 7. So the first equation becomes 2x + 3y + 7z = 7. For the second equation, 14t - 14 becomes 14(0) - 14, which is -14. So the second equation becomes 5x + 3y + 4z = -14. Now we have a new system:
2x + 3y + 7z = 7
5x + 3y + 4z = -14
This system looks a lot like the one we were working with before, but now the right-hand sides are just constants instead of expressions involving 't'. This often makes the system easier to solve. We can use the same techniques we discussed earlier – elimination, substitution, or matrices – to find the values of x, y, and z that satisfy these equations. Remember that earlier, we used elimination to get the equation x - z = -7. This equation is still valid when t=0, so we can use it along with our new equations to solve for the variables. Having this specific case (t=0) is helpful because it removes a degree of freedom from the system. In the general case, the solutions might depend on the value of 't', meaning there could be infinitely many solutions. But when we fix 't' to a specific value, like 0, we often narrow down the possibilities and make it easier to find concrete solutions. It's like taking a snapshot of the system at a particular moment in time.
Adding the Condition x-y=0
Now, let's throw another wrench into the works! We're going to add the condition x-y=0. This is an extra piece of information that can potentially change the solutions we find. When we add conditions like this, we're essentially narrowing down the set of possible solutions even further. Think of it like this: the original system of equations represents a set of solutions. The condition x-y=0 represents another set of solutions. The solutions that satisfy both the system and the condition are the ones that lie in the intersection of these two sets. So, what does x-y=0 actually mean? It simply means that x is equal to y. In other words, the x and y coordinates have to be the same. This gives us a direct relationship between two of our variables. To see how this affects our solutions, we need to incorporate this condition into our system of equations. We can do this by using substitution. Since x=y, we can replace every 'y' in our equations with 'x' (or vice versa). This will give us a new system with only two variables, x and z, which should be easier to solve. Let's take our system when t=0:
2x + 3y + 7z = 7
5x + 3y + 4z = -14
And substitute y with x:
2x + 3x + 7z = 7
5x + 3x + 4z = -14
Now we can simplify these equations:
5x + 7z = 7
8x + 4z = -14
We also have our earlier equation x - z = -7, which we can rewrite as x = z - 7. We can use this equation along with our new system to solve for x and z. Once we have x and z, we can easily find y since y=x. Adding the condition x-y=0 has transformed our problem. It's given us an extra constraint that helps us pinpoint the specific solutions that meet all the requirements. This is a common theme in math and science – adding constraints often leads to more specific and meaningful solutions.
Finding the Solution with x-y=0
Okay, guys, let's get down to brass tacks and find the solution with x-y=0. We've simplified our system to these equations:
5x + 7z = 7
8x + 4z = -14
And we also know that x = z - 7. We can use this last equation to substitute for x in the other two equations. Let's start with the first equation, 5x + 7z = 7. Replacing x with (z - 7), we get:
5(z - 7) + 7z = 7
Now, let's distribute the 5 and simplify:
5z - 35 + 7z = 7
12z - 35 = 7
Add 35 to both sides:
12z = 42
And finally, divide by 12:
z = 42/12 = 7/2
So we've found that z = 7/2. Awesome! Now we can use x = z - 7 to find x:
x = (7/2) - 7 = (7/2) - (14/2) = -7/2
So x = -7/2. And since x = y, we know that y = -7/2 as well. We've found our solution! When t=0 and x-y=0, the solution to the system is:
x = -7/2
y = -7/2
z = 7/2
This is a specific, unique solution that satisfies all the conditions we've imposed. It's pretty cool how adding the condition x-y=0 helped us narrow down the possibilities and pinpoint this exact solution. We started with a general system of equations, then looked at the specific case when t=0, and finally added the constraint x-y=0. Each step brought us closer to a concrete answer. This whole process highlights the power of using different techniques and conditions to solve mathematical problems. By combining elimination, substitution, and careful consideration of special cases, we were able to unravel this system and find the solution. So, next time you're faced with a system of equations, remember these tools and approaches – they can be your best friends in the world of math! This solution gives a complete overview of how systems of equations can be solved by applying different constraints and simplification methods.
Conclusion
Wrapping things up, guys, we've journeyed through solving a system of equations, explored the scenario when t=0, and then added the condition x-y=0. We used elimination and substitution to simplify the equations and pinpoint the solution. This whole process highlights how different constraints and conditions can shape the solutions to mathematical problems. Systems of equations are a fundamental tool in many fields, and understanding how to solve them is a valuable skill. By mastering techniques like elimination, substitution, and considering specific cases, you can tackle a wide range of problems in math, science, and beyond. Remember, math isn't just about finding the right answer – it's about the journey of problem-solving and the insights you gain along the way. So keep exploring, keep questioning, and keep solving!