Solving Systems: Find X, Y, And Z!

by Andrew McMorgan 35 views

Hey Plastik Magazine readers! Ever stared at a system of equations and felt a little lost? Don't worry, we've all been there! But fear not, because today we're diving deep into the world of systems of equations, specifically focusing on how to find the values of x, y, and z when presented with a set of three equations. We'll break down the process step-by-step, making it easy to understand even if you're not a math whiz. We'll be using methods like elimination and substitution to crack these mathematical puzzles. Let's get started and unravel this together, shall we? You'll be surprised at how logical and manageable it can be! Think of it like a fun puzzle – and who doesn't love a good puzzle?

Understanding the System of Equations

Alright, let's get down to business. A system of equations, in this case, consists of three equations, each with three variables: x, y, and z. Our goal is to find the unique values for each of these variables that satisfy all three equations simultaneously. In our specific problem, we have the following system:

  1. 8x + 3y + 6z = 43
  2. -3x + 5y + 2z = 32
  3. 5x - 2y + 5z = 24

Each equation represents a plane in 3D space, and the solution to the system is the point where all three planes intersect. Think of it like this: each equation is a clue, and we need to use all three clues to pinpoint the exact location of our solution. The methods we'll use are designed to systematically eliminate variables until we can isolate each one and solve for its value. The key is to be organized and methodical. Double-check your calculations at each step to avoid errors. Let's make sure we get this right, guys! Are you ready to dive in and learn how to solve this? Let's take it one step at a time, we can totally do this! And trust me, once you get the hang of it, it's quite satisfying to solve these!

The Elimination Method: A Powerful Tool

One of the most effective methods for solving systems of equations is the elimination method. The core idea is to manipulate the equations (by multiplying them by constants) so that when you add or subtract them, one of the variables cancels out. This reduces the problem to a system with fewer variables, making it easier to solve. It's like strategically canceling out terms to simplify the problem, kind of like a magic trick, but with math! It’s all about finding clever ways to combine the equations to eliminate variables. It will be fun, believe me!

Let's apply the elimination method to our system of equations. First, we'll aim to eliminate z. We can achieve this by multiplying equations (2) by -3 and adding it to equation (1). So, this gives us:

  • Multiply equation (2) by -3: -3 * (-3x + 5y + 2z) = -3 * 32 which simplifies to 9x - 15y - 6z = -96
  • Add the modified equation (2) to equation (1): (8x + 3y + 6z) + (9x - 15y - 6z) = 43 + (-96). This simplifies to 17x - 12y = -53. Let's call this new equation (4).

Now, let's eliminate z again, this time using equations (2) and (3). We'll multiply equation (2) by -5/2 and add it to equation (3). We need to do this again to eliminate z completely from two new equations.

  • Multiply equation (2) by -5/2: -5/2 * (-3x + 5y + 2z) = -5/2 * 32 which simplifies to (15/2)x - (25/2)y - 5z = -80
  • Add the modified equation (2) to equation (3): (5x - 2y + 5z) + ((15/2)x - (25/2)y - 5z) = 24 + (-80). This simplifies to (25/2)x - (29/2)y = -56. Multiply both sides by 2 to get 25x - 29y = -112. Let's call this new equation (5).

Awesome, now we have two equations with only two variables! We've made great progress. We're getting closer to solving this step by step. Are you following along, guys? Don't hesitate to reread a step if you need to; it's all about understanding the process! Let's get to the next step.

Solving for x and y

Now we have two new equations:

  1. 17x - 12y = -53
  2. 25x - 29y = -112

We can use the elimination method again, but this time to solve for either x or y. Let's solve for x. To do this, we can multiply equation (4) by 29 and equation (5) by -12. Then, adding the resulting equations together will eliminate y. Remember to be careful with the signs and coefficients! It's super important. Are you ready? Let's go!

  • Multiply equation (4) by 29: 29 * (17x - 12y) = 29 * (-53) which simplifies to 493x - 348y = -1537
  • Multiply equation (5) by -12: -12 * (25x - 29y) = -12 * (-112) which simplifies to -300x + 348y = 1344
  • Add the two equations: (493x - 348y) + (-300x + 348y) = -1537 + 1344. This simplifies to 193x = -193

Now, we can solve for x: x = -193 / 193, so x = -1.

We have found the value for x! Now that we know x, let's substitute it into either equation (4) or (5) to find y. Let's use equation (4): 17x - 12y = -53. Substituting x = -1, we get: 17*(-1) - 12y = -53, which simplifies to -17 - 12y = -53. Add 17 to both sides: -12y = -36. Dividing by -12 gives us y = 3. See? We're on a roll now! We're doing great, guys!

Finding the Value of z

We're in the home stretch now, guys! We've found the values for x and y, so now it's time to find z. We can substitute the values of x and y into any of the original three equations. Let's use the first equation: 8x + 3y + 6z = 43. Substituting x = -1 and y = 3, we get: 8*(-1) + 3*(3) + 6z = 43, which simplifies to -8 + 9 + 6z = 43, and then to 1 + 6z = 43. Subtract 1 from both sides: 6z = 42. Finally, divide by 6: z = 7.

And there you have it! We've successfully solved the system of equations and found the values for all three variables.

  • x = -1
  • y = 3
  • z = 7

Congratulations, we did it! Awesome job, everyone. Give yourselves a pat on the back! Solving these systems might seem daunting at first, but with a systematic approach and practice, you can conquer any equation. Remember to stay organized, check your work, and don't be afraid to ask for help if you need it. Now go out there and amaze yourselves, you amazing math wizards!