Solving Systems Of Equations: A Step-by-Step Guide
Hey math enthusiasts! Ever find yourself staring at a system of equations and feeling totally lost? Don't worry, you're not alone! Systems of equations can seem intimidating, but with the right approach, they're totally solvable. In this guide, we'll break down a specific system of equations and walk through the steps to find the solution. So, let's dive in and conquer those equations together!
Understanding Systems of Equations
Before we jump into solving, let's quickly recap what a system of equations actually is. A system of equations is simply a set of two or more equations that share the same variables. Our goal is to find values for these variables that satisfy all equations simultaneously. Think of it like finding the sweet spot where all the equations agree. In our case, we're dealing with a system of three linear equations with three variables (x, y, and z). These types of systems pop up all over the place in math, science, and even real-life problem-solving, so mastering them is a super valuable skill.
Linear equations are equations where the variables are only raised to the first power (no exponents or funky functions), and they graph as straight lines. When you have multiple lines, the solution to the system is the point (or points) where the lines intersect. For three variables, we're dealing with planes instead of lines, but the idea is the same: we're looking for the point where all the planes meet. This intersection point represents the values of x, y, and z that work in all three equations. There are several methods for solving systems of equations, such as substitution, elimination, and matrix methods. We'll be using the elimination method in this guide, but it's always good to be familiar with different approaches. The beauty of math is that there's often more than one way to reach the same destination! So, let's get ready to solve a system of equations step by step. This method is powerful because it allows us to strategically eliminate variables, making the system simpler to solve. By combining equations in a clever way, we can reduce the number of unknowns and eventually isolate the variables we're looking for. So, let's get started and see how this works in practice!
The System We're Tackling
Okay, let's take a look at the specific system of equations we're going to solve today:
\begin{cases}
x + y + z = 6 \\
x - y + z = 8 \\
x + y - z = 0
\end{cases}
This system consists of three equations, each with the variables x, y, and z. Our mission, should we choose to accept it (and we do!), is to find the values of x, y, and z that make all three equations true at the same time. Looks a bit daunting, right? But fear not! We're going to break it down into manageable steps, and you'll see that it's not as scary as it looks. First thing's first, let's label these equations to keep things organized. We'll call the first equation (1), the second equation (2), and the third equation (3). This will make it easier to refer to them as we work through the solution. Now, before we start crunching numbers, let's take a moment to strategize. What's the best way to approach this system? Notice anything interesting about the equations? Maybe you spot some variables that have opposite signs or coefficients. This is a clue that the elimination method might be a good choice. Remember, the goal of the elimination method is to eliminate one variable at a time by adding or subtracting equations. By strategically choosing which equations to combine, we can whittle down the system until we're left with a single equation in a single variable. Then, it's smooth sailing from there! So, with our strategy in mind, let's roll up our sleeves and start solving. The first step is to decide which variable we want to eliminate first. Any ideas? Keep reading, and we'll figure it out together!
Step 1: Eliminating 'y'
Alright, let's get down to business! Looking at our system of equations, we can see that the 'y' variable has both positive and negative coefficients. This is fantastic news because it means we can easily eliminate 'y' by adding equations together. Remember, the goal of elimination is to make the coefficients of one variable opposites so that they cancel out when you add the equations. In this case, the 'y' terms in equations (1) and (2) are already set up perfectly for elimination: one is +y and the other is -y. So, let's add equation (1) and equation (2) together. When we do this, the 'y' terms will magically disappear, leaving us with an equation in just x and z. This is a huge step forward because we've reduced the number of variables in the equation. But wait, there's more! We can also eliminate 'y' by adding equation (1) and equation (3) together. Again, the 'y' terms will cancel out, giving us another equation in x and z. Now we have two equations with the same two variables, which is a much simpler system to solve. The beauty of this method is that we're systematically reducing the complexity of the problem. By eliminating one variable at a time, we're gradually working our way towards the solution. It's like peeling back the layers of an onion, one at a time, until we reach the core. So, let's grab a pen and paper and actually perform the addition. Make sure to add the corresponding terms together carefully, and don't forget the constants on the right-hand side of the equations. Once we've done the addition, we'll have two new equations that are much easier to work with. These new equations will be the key to unlocking the values of x and z. Are you ready to see how it's done? Let's go!
Adding equations (1) and (2):
(x + y + z) + (x - y + z) = 6 + 8
Simplifies to:
2x + 2z = 14
We can further simplify this by dividing both sides by 2:
x + z = 7 (4)
Now, let's add equations (1) and (3):
(x + y + z) + (x + y - z) = 6 + 0
Simplifies to:
2x + 2y = 6
Dividing both sides by 2:
x + y = 3 (5)
Step 2: Solving for 'x' and 'z'
Awesome! We've successfully eliminated 'y' and now have a simpler system of two equations: equation (4) which is x + z = 7, and equation (5) which is x + y = 3. However, we still have three variables (x, y, and z), so we need to eliminate one more variable to solve for the remaining ones. Let's focus on equations (4) and the modified version of equation (5) (we'll get to that in a sec). These equations involve only x and z, which is exactly what we want. To eliminate another variable, we need to manipulate these equations so that the coefficients of either 'x' or 'z' are opposites. Looking at equation (4) and equation (5), we see that the 'x' terms already have the same coefficient (1). This means we can eliminate 'x' by subtracting one equation from the other. Which equation should we subtract from which? Well, it doesn't really matter, but it's often a good idea to subtract the equation with the smaller constant term from the equation with the larger constant term. This can help avoid negative numbers, which can sometimes lead to errors. So, in this case, let's subtract equation (5) from equation (4). This will give us a new equation with only 'z', which we can then easily solve. Once we have the value of 'z', we can substitute it back into either equation (4) or equation (5) to find the value of 'x'. It's like a domino effect: once we knock down one domino (solve for one variable), the rest will fall into place. So, let's get subtracting and see what we get! Remember to subtract the corresponding terms carefully, and keep track of the signs. With a little bit of algebraic maneuvering, we'll have the values of 'x' and 'z' in no time. Are you ready for the next step? Let's do it!
Before we subtract, let's rewrite equation (5) in terms of x and y:
x + y = 3
Oops! It seems we made a slight detour in our previous steps. Equation (5) actually involves x and y, not x and z. No worries, we can still use it, but we need to go back and find another equation that relates x and z. Remember equation (4), x + z = 7? That's the one we need! So, let's stick with equation (4) and try a different approach. Instead of subtracting equations directly, let's use substitution. We can solve equation (4) for one variable (say, x) and then substitute that expression into equation (5). This will give us an equation with only one variable (y), which we can easily solve. Once we have 'y', we can substitute it back into either equation to find 'x'. It's like a puzzle, and we're just rearranging the pieces until they fit together. So, let's solve equation (4) for x:
x = 7 - z
Now, substitute this expression for x into equation (5):
(7 - z) + y = 3
Step 3: Solving for 'y'
Okay, we've made some progress! We've substituted the expression for 'x' (which is 7 - z) into equation (5), and now we have a new equation: (7 - z) + y = 3. This equation has three variables (y, z), but we also have other equations in our system that relate these variables. Remember, our goal is to isolate one variable at a time, so let's see if we can use this new equation to do that. Looking at the equation, we can rearrange it to solve for 'y'. This means getting 'y' all by itself on one side of the equation. To do this, we need to get rid of the other terms on the same side as 'y'. In this case, we have 7 and -z. So, how do we get rid of them? We use inverse operations! To get rid of 7, we subtract 7 from both sides of the equation. And to get rid of -z, we add z to both sides. Remember, whatever we do to one side of the equation, we have to do to the other side to keep it balanced. It's like a seesaw: if you add weight to one side, you have to add the same weight to the other side to keep it level. Once we've done these operations, we'll have 'y' all by itself on one side, and an expression involving 'z' on the other side. This will give us the value of 'y' in terms of 'z'. But we're not done yet! We still need to find the actual numerical value of 'y'. To do this, we'll need to find the value of 'z' first. Don't worry, we're getting closer! We've already eliminated one variable ('y' in the first step), and now we're about to find the value of another variable. It's like a scavenger hunt: each step brings us closer to the final treasure. So, let's rearrange the equation and see what we get! Are you ready to find 'y'? Let's go!
Rearranging the equation (7 - z) + y = 3 to solve for y:
y = 3 - 7 + z
Simplifies to:
y = z - 4 (6)
Step 4: Solving for 'z'
Excellent! We've now found an expression for 'y' in terms of 'z': y = z - 4 (equation 6). This is a major step forward because it means we've reduced the number of unknowns. We're getting closer and closer to cracking this system! But we still need to find the actual value of 'z'. To do this, we need to use another equation that involves 'z'. Remember those original equations we started with? It's time to revisit them! We can substitute our expression for 'y' (z - 4) into one of the original equations, and this will give us an equation with only 'z' as the variable. Which equation should we choose? Well, it doesn't really matter, but sometimes one equation might look easier to work with than another. In this case, let's choose equation (1): x + y + z = 6. It looks relatively simple, and it has all three variables, so it's a good candidate. Now, before we substitute, we need to do one more thing. We also need to substitute our expression for 'x' (7 - z) into this equation. Remember, we found this expression back in Step 2 when we were solving for 'x' and 'z'. So, we're substituting two expressions into one equation: y = z - 4 and x = 7 - z. This might seem like a lot of substitutions, but it's all part of the process of elimination. By substituting, we're effectively getting rid of the 'x' and 'y' variables, leaving us with an equation that we can solve for 'z'. It's like a carefully choreographed dance: each substitution is a step that leads us closer to the final solution. So, let's grab our substitution shoes and get ready to dance! We're about to find the value of 'z', and once we have that, the rest will fall into place. Are you excited? Let's do it!
Substitute x = 7 - z and y = z - 4 into equation (1):
(7 - z) + (z - 4) + z = 6
Step 5: Calculating 'z'
Alright, we've made our substitutions, and now we have the equation (7 - z) + (z - 4) + z = 6. Take a deep breath, because we're about to simplify this and finally solve for 'z'! Remember, the key to simplifying equations is to combine like terms. This means grouping together the terms that have the same variable (in this case, 'z') and the terms that are just numbers (the constants). So, let's look at our equation and identify the 'z' terms and the constant terms. We have -z, +z, and +z as the 'z' terms, and we have 7 and -4 as the constant terms. Now, let's combine them! When we combine -z and +z, they cancel each other out, leaving us with just +z. And when we combine 7 and -4, we get 3. So, our equation simplifies to z + 3 = 6. See? It's already looking much simpler! Now, to solve for 'z', we need to get 'z' all by itself on one side of the equation. This means getting rid of the +3 on the same side as 'z'. How do we do that? We use the inverse operation! The inverse operation of addition is subtraction, so we subtract 3 from both sides of the equation. Remember, whatever we do to one side, we have to do to the other side to keep the equation balanced. When we subtract 3 from both sides, the +3 on the left side cancels out, leaving us with just 'z'. And on the right side, we have 6 - 3, which is 3. So, we've found it! The value of 'z' is 3. Hallelujah! We've cracked one of the variables! This is a huge accomplishment, and it's time to celebrate our progress. But we're not quite done yet. We still need to find the values of 'x' and 'y'. But don't worry, now that we know 'z', finding the other variables will be much easier. We can use our values which we already found to easily calculate for 'x' and 'y'. Are you ready to finish this? Let's do it!
Simplifying the equation:
7 - z + z - 4 + z = 6
z + 3 = 6
Solving for z:
z = 6 - 3
z = 3
Step 6: Finding 'x' and 'y'
YES! We've nailed it! We know that z = 3. Give yourself a pat on the back, because that's a major step! Now, the finish line is in sight. We just need to find the values of x and y, and we'll have the complete solution to our system of equations. Lucky for us, we've already done most of the hard work. We have expressions for x and y in terms of z, which we found in previous steps. Remember back in Step 2, we found that x = 7 - z? And in Step 3, we found that y = z - 4? Now, all we need to do is plug in our value for z (which is 3) into these expressions, and we'll instantly have the values of x and y. It's like a mathematical magic trick! We've transformed a complicated system of equations into a simple substitution problem. This is the power of the elimination method: it breaks down a complex problem into smaller, more manageable steps. So, let's grab our trusty substitution hats again and get ready to plug in z = 3. We'll start with the expression for x: x = 7 - z. We substitute 3 for z, and we get x = 7 - 3. Simple subtraction tells us that x = 4. Boom! We've found x! Now, let's move on to y. We have the expression y = z - 4. We substitute 3 for z, and we get y = 3 - 4. This time, we're subtracting a larger number from a smaller number, so we'll get a negative result. 3 - 4 is -1, so y = -1. Double boom! We've found y! We now know the values of all three variables: x = 4, y = -1, and z = 3. We've solved the system of equations! Are you feeling like a math superstar? You should be! We took a challenging problem and broke it down step by step, and now we have the solution. Let's write it down clearly so we can admire our handiwork. Are you ready to see the final answer? Let's go!
Substitute z = 3 into x = 7 - z:
x = 7 - 3
x = 4
Substitute z = 3 into y = z - 4:
y = 3 - 4
y = -1
The Solution!
Drumroll, please! After all our hard work, we've finally arrived at the solution to the system of equations. We've found the values of x, y, and z that satisfy all three equations simultaneously. And the answer is:
(x, y, z) = (4, -1, 3)
Isn't that satisfying? It's like completing a puzzle and seeing all the pieces fit perfectly together. This solution means that if we substitute x = 4, y = -1, and z = 3 into any of the original equations, the equation will be true. You can even try it out yourself to double-check our answer! But before we celebrate too much, let's take a moment to reflect on the process we used to get here. We started with a seemingly complicated system of equations, and we systematically broke it down into smaller, more manageable steps. We used the elimination method to get rid of variables one at a time, and we used substitution to find the values of the remaining variables. We encountered a few detours along the way, but we didn't give up! We adjusted our approach and kept moving forward. This is what problem-solving is all about: being persistent, being flexible, and being willing to try different strategies. So, the next time you encounter a system of equations, remember this journey. Remember the steps we took, and remember that you have the power to solve it! You've learned a valuable skill today, and you can apply it to all sorts of problems, both in math and in real life. Give yourself a huge round of applause! You've earned it! And now, let's admire our beautiful solution one more time: (4, -1, 3). It's a testament to our hard work and our mathematical prowess. Congratulations!
Choosing the Correct Option
Now that we've meticulously solved the system of equations, we know the solution is (x, y, z) = (4, -1, 3). Let's take a look at the options provided and see which one matches our solution. This is the final step in the process, and it's important to make sure we choose the correct answer. Sometimes, test questions are designed to trick you, so it's always a good idea to double-check your work and make sure you're selecting the right option. We've already done the hard part of solving the problem, so this should be the easy part! But it's still important to pay attention to detail. Let's go through the options one by one and compare them to our solution. Option A is (3, -1, 4). Notice that the values are the same as our solution, but they're in a different order. Remember, the order matters! The first number is the value of x, the second number is the value of y, and the third number is the value of z. So, (3, -1, 4) is not the same as (4, -1, 3). Option B is (4, 1, 3). This option has the correct values for x and z, but the value for y is incorrect. We found that y = -1, not 1. So, option B is also incorrect. Option C is (4, -1, 3). Bingo! This option matches our solution perfectly! The values are in the correct order, and they're the same values we found for x, y, and z. So, option C is the correct answer. But just to be sure, let's take a quick look at option D. Option D is (3, 1, 4). This option has the same values as option A, just in a different order. And we already know that option A is incorrect, so option D is also incorrect. So, we've confirmed that option C is the correct answer. We've solved the system of equations, and we've identified the correct option. We're math champions! Give yourself one more round of applause, because you deserve it! You've successfully navigated a system of equations, and you've learned a valuable skill that will serve you well in the future. Congratulations!
The correct answer is:
C. (4, -1, 3)
Final Thoughts
Woohoo! We did it! We successfully solved a system of three equations with three variables. Give yourselves a huge pat on the back, guys! You've proven that you can tackle even the trickiest math problems with a little bit of strategy and a whole lot of perseverance. Solving systems of equations might seem like a purely mathematical exercise, but it's actually a valuable skill that can be applied in many different areas of life. From science and engineering to economics and finance, systems of equations pop up everywhere. Being able to solve them effectively can give you a real edge in these fields. But more than that, the process of solving systems of equations teaches you valuable problem-solving skills that can be applied to any challenge you face. It teaches you how to break down a complex problem into smaller, more manageable steps. It teaches you how to think strategically and plan your approach. And it teaches you the importance of persistence and not giving up when things get tough. So, the next time you're faced with a difficult problem, remember the lessons you've learned here. Remember the steps we took to solve this system of equations. And remember that you have the power to find the solution, one step at a time. Keep practicing, keep learning, and keep challenging yourselves. You never know what amazing things you'll be able to accomplish! And always remember, math can be fun! It's like a puzzle, and the satisfaction of finding the solution is truly rewarding. So, embrace the challenge, enjoy the journey, and keep on solving!