Solving Systems Of Equations: A Step-by-Step Guide

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Hey guys! Let's dive into the fascinating world of solving systems of equations. You know, those tricky problems where you have multiple equations and multiple unknowns? Today, we're going to tackle one such system step-by-step, making sure everyone understands the process. Our main goal here is to provide a comprehensive guide on how to approach and solve these types of problems. We'll focus on a specific example, but the principles we discuss can be applied to a wide range of systems of equations. So, grab your pencils, and let's get started!

The System of Equations

Before we jump into the solution, let's first clearly state the system of equations we'll be working with. This is crucial because understanding the problem is half the battle, right? We have three equations and three unknowns (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 satisfy all three equations simultaneously. This is where the magic of algebra happens!

The system is as follows:

  1. x - y + z = 5
  2. -x + 4y + 2z = 10
  3. -x + 3y - 5z = -7

Analyzing the equations is our first step. Notice anything interesting? Maybe you see that the 'x' terms in the second and third equations have opposite signs compared to the first. This is a clue! It suggests that we might be able to use elimination to simplify the system. Elimination is a powerful technique where we add or subtract equations to get rid of one variable, making the system easier to solve. We'll delve into this shortly, but for now, just keep this observation in mind. Remember, being observant and looking for patterns is key to becoming a master problem-solver in mathematics!

Step 1: Elimination of 'x'

The elimination method is our go-to strategy here. Remember how we noticed those 'x' terms with opposite signs? Well, that's our ticket to simplifying things. Our plan is to add the first equation to the second and third equations. This will eliminate 'x' from those equations, leaving us with a smaller system involving only 'y' and 'z'. Think of it as strategically knocking out one variable at a time, like a mathematical game of dominoes!

Let's do it:

  • Adding equation 1 to equation 2: (x - y + z) + (-x + 4y + 2z) = 5 + 10 This simplifies to: 3y + 3z = 15. We can further simplify this by dividing both sides by 3, giving us: y + z = 5. Let's call this equation 4. This is a crucial new equation.
  • Adding equation 1 to equation 3: (x - y + z) + (-x + 3y - 5z) = 5 + (-7) This simplifies to: 2y - 4z = -2. Again, we can simplify by dividing both sides by 2, resulting in: y - 2z = -1. Let's call this equation 5. Another important equation in our journey.

Now, look what we've done! We've transformed our original system of three equations into a new, simpler system of two equations (equations 4 and 5) with only two unknowns ('y' and 'z'). This is progress! We're one step closer to cracking the code and finding the values of x, y, and z. The power of elimination is truly on display here. We've effectively reduced the complexity of the problem, making it much more manageable.

Step 2: Solving for 'y' and 'z'

Now that we've eliminated 'x', we're left with a system of two equations (equation 4 and equation 5) involving 'y' and 'z'. This is a much more manageable situation! We can use either substitution or elimination again to solve for these variables. Let's stick with elimination, as it worked so well for us before. We're building momentum here, guys!

Our equations are:

  1. y + z = 5
  2. y - 2z = -1

Notice that the 'y' terms have the same coefficient. This means we can easily eliminate 'y' by subtracting equation 5 from equation 4. Think of it as carefully targeting the 'y' variable for removal.

Let's subtract:

(y + z) - (y - 2z) = 5 - (-1) This simplifies to: 3z = 6

Now, divide both sides by 3 to isolate 'z': z = 2

Fantastic! We've found the value of z! This is a major breakthrough. Now that we know z = 2, we can substitute this value back into either equation 4 or 5 to solve for 'y'. Let's use equation 4, as it looks a bit simpler.

Substitute z = 2 into equation 4: y + 2 = 5 Subtract 2 from both sides: y = 3

We've found the value of 'y' as well! This is amazing progress. We now know that y = 3 and z = 2. We're just one step away from solving the entire system. We've conquered two-thirds of the problem, and the final piece of the puzzle is within our grasp. This feeling of accomplishment is what makes math so rewarding, isn't it?

Step 3: Solving for 'x'

We're in the home stretch now! We've successfully found the values of 'y' and 'z' (y = 3 and z = 2). All that remains is to find 'x'. To do this, we can substitute the values of 'y' and 'z' into any of the original three equations. Let's choose the first equation (x - y + z = 5) because it looks the simplest. Remember, efficiency is key in problem-solving!

Substitute y = 3 and z = 2 into equation 1: x - 3 + 2 = 5 Simplify: x - 1 = 5 Add 1 to both sides: x = 6

We've done it! We've found the value of 'x'! We now have all the pieces of the puzzle: x = 6, y = 3, and z = 2. This is the solution to our system of equations. The feeling of finally cracking a tough problem is truly satisfying.

The Solution

So, the solution to the system of equations is:

x = 6 y = 3 z = 2

We can write this as an ordered triple: (6, 3, 2). This represents the point in three-dimensional space where all three planes (represented by the equations) intersect. Cool, right?

But wait! Before we celebrate too much, there's one crucial step we need to take: verification. We need to make sure our solution actually works. This means plugging the values of x, y, and z back into the original equations to see if they hold true. This is like a final quality check, ensuring that our solution is rock solid.

Let's verify:

  • Equation 1: 6 - 3 + 2 = 5 (Correct!)
  • Equation 2: -6 + 4(3) + 2(2) = -6 + 12 + 4 = 10 (Correct!)
  • Equation 3: -6 + 3(3) - 5(2) = -6 + 9 - 10 = -7 (Correct!)

Our solution checks out! This is the ultimate confirmation that we've solved the system correctly. We can now confidently say that (6, 3, 2) is the solution to the given system of equations.

Key Takeaways

Let's recap the key strategies we used to solve this system of equations. These are valuable techniques that you can apply to other similar problems:

  1. Elimination Method: We strategically added and subtracted equations to eliminate variables, simplifying the system step-by-step.
  2. Substitution: Once we found the value of one variable, we substituted it back into other equations to solve for the remaining variables.
  3. Verification: We always verified our solution by plugging the values back into the original equations to ensure accuracy.

These are powerful tools in your mathematical arsenal. Mastering these techniques will enable you to tackle a wide range of systems of equations with confidence. Remember, practice makes perfect! The more you solve these types of problems, the more comfortable and proficient you'll become.

Conclusion

Solving systems of equations can seem daunting at first, but by breaking it down into smaller, manageable steps, it becomes much more approachable. We've walked through a detailed example, highlighting the elimination method, substitution, and the crucial step of verification. Remember, the key is to be organized, methodical, and persistent. Don't be afraid to experiment and try different approaches. And most importantly, have fun with it! Math is a beautiful and powerful tool, and solving problems like this can be incredibly rewarding.

So, go forth and conquer those systems of equations! You've got this! And remember, if you ever get stuck, there are plenty of resources available online and in textbooks to help you out. Keep practicing, keep learning, and keep exploring the fascinating world of mathematics! You're all math whizzes in the making!