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

Hey guys! Ever stumbled upon a system of equations that looks like it was written in a different language? Don't worry, we've all been there. These things can seem intimidating, but with the right approach, you can crack them like a pro. Today, we're going to break down a specific system of equations and show you exactly how to solve it. So, grab your pencils, and let's dive in!

The Challenge: Our System of Equations

Before we get into the nitty-gritty, let's take a look at the system we're tackling. It's a set of three equations with four variables (x, y, z, and t):

-0.2x + 0.3y + 0.4z - t = 4.4
2.2x + 1.1y - 4.7z + 2t = 9.5
9.2y + 0.5z - 3.4t = -1.7

Okay, I know what you're thinking: "Whoa, that looks complicated!" And you're not wrong, it does have a few more variables than we might be used to seeing. But don't let that scare you off. We're going to take it one step at a time, and you'll see that it's totally manageable. The key to these systems of equations is to systematically eliminate variables until we can isolate one and solve for it. Then, we can back-substitute to find the values of the others. This is where the fun begins, as we use algebraic manipulations to unravel this mathematical puzzle.

We need to consider what makes this system of equations unique. It's not just about finding any solution; it's about understanding the relationship between the equations and the variables. This involves careful observation and strategic thinking. We'll be looking for patterns, opportunities for simplification, and the most efficient path to a solution. Solving this system isn't just about getting the right answers; it's about developing a deeper understanding of algebraic principles. So, let's get started and transform this complex problem into a series of manageable steps. Remember, every great mathematical journey begins with a single step, and we're about to take that step together. Are you ready to unlock the secrets of this equation system?

Step 1: Elimination Strategy

Our main goal here is to reduce the complexity of the system. We're going to aim to eliminate one variable at a time, making the equations easier to solve. A good strategy is to look for variables that have coefficients that are easy to work with or can be easily manipulated to match. In our case, let's focus on eliminating 'x' from the second equation. Notice that the first equation has a -0.2x term, and the second equation has a 2.2x term. If we multiply the first equation by 11, the 'x' term will become -2.2x, which is the exact opposite of the 2.2x in the second equation. This sets us up perfectly for elimination through addition.

So, let's multiply the first equation by 11:

11 * (-0.2x + 0.3y + 0.4z - t) = 11 * 4.4
-2.2x + 3.3y + 4.4z - 11t = 48.4

Now we have a modified first equation that's ready to be combined with the second equation. This process of strategic manipulation is crucial in solving systems of equations. It's not just about randomly adding or subtracting equations; it's about thinking ahead and choosing operations that will lead to the simplest possible form. By focusing on 'x' and using multiplication to align the coefficients, we're setting the stage for a clean elimination. This step is a perfect example of how a little foresight can save a lot of effort later on. The idea here is to carefully plan our moves, like a chess player, to gain an advantage in the solving process. So, with our modified equation in hand, let's move on to the next step and see how this elimination strategy pays off!

Step 2: Eliminating 'x'

Now that we've multiplied the first equation by 11, we're all set to eliminate 'x' from the second equation. We'll do this by adding the modified first equation to the second equation. Remember, the goal is to make the 'x' terms cancel each other out, leaving us with an equation that has one less variable. Let's line up the equations and perform the addition:

-2.2x + 3.3y + 4.4z - 11t = 48.4  (Modified first equation)
 2.2x + 1.1y - 4.7z + 2t = 9.5   (Second equation)
-------------------------------------
 0x + 4.4y - 0.3z - 9t = 57.9   (Result of addition)

As you can see, the 'x' terms have canceled out beautifully, leaving us with a new equation that involves only 'y', 'z', and 't'. This is a huge step forward! We've effectively reduced the complexity of the system by eliminating one variable from one of the equations. This process highlights the power of elimination in simplifying complex problems. By strategically combining equations, we can peel away layers of complexity and get closer to a solution. The resulting equation, 4.4y - 0.3z - 9t = 57.9, is a key piece of the puzzle. It connects the variables 'y', 'z', and 't' in a way that will help us further reduce the system. This step demonstrates that solving systems of equations is not just about applying formulas; it's about using creative algebraic techniques to simplify the problem. So, with this new equation in our arsenal, we're ready to tackle the next challenge and continue our journey towards solving the system!

Step 3: Rewriting the System

Okay, we've made some solid progress! We eliminated 'x' from the second equation and got a new equation in terms of 'y', 'z', and 't'. Now, let's rewrite our system to reflect this change. This will give us a clearer picture of what we have to work with and help us plan our next move. Our updated system of equations looks like this:

-0.2x + 0.3y + 0.4z - t = 4.4
 4.4y - 0.3z - 9t = 57.9  (New equation)
 9.2y + 0.5z - 3.4t = -1.7

Notice how the system now has a slightly different structure. The second equation no longer contains 'x', which means we've effectively reduced the number of variables in that part of the system. This is a critical step in the solving process because it allows us to focus on the remaining variables and their relationships. Rewriting the system in this way is like organizing your workspace before tackling a big project. It helps you see the overall picture and identify the most efficient way to proceed. By keeping the system organized, we can avoid confusion and ensure that we're making progress towards our goal. This step also highlights the importance of strategic thinking in problem-solving. It's not just about performing calculations; it's about understanding the structure of the problem and making informed decisions about how to approach it. So, with our system neatly rewritten, let's move on to the next phase and continue our quest for a solution!

Step 4: Eliminating Another Variable

Now, let's continue our strategy of elimination. Looking at our updated system, we have two equations (the second and third) that involve only 'y', 'z', and 't'. This gives us a great opportunity to eliminate another variable. Let's focus on eliminating 'z' from these two equations. To do this, we need to find a way to make the coefficients of 'z' in the two equations opposites of each other. The second equation has a -0.3z term, and the third equation has a 0.5z term. A common multiple of 0.3 and 0.5 is 1.5. So, we can multiply the second equation by 5 and the third equation by 3 to make the 'z' coefficients -1.5 and 1.5, respectively.

Let's perform these multiplications:

Multiply the second equation by 5:

5 * (4.4y - 0.3z - 9t) = 5 * 57.9
22y - 1.5z - 45t = 289.5

Multiply the third equation by 3:

3 * (9.2y + 0.5z - 3.4t) = 3 * (-1.7)
27.6y + 1.5z - 10.2t = -5.1

We've now transformed the two equations so that the 'z' terms are ready for elimination. This process of strategic multiplication is a key technique in solving systems of equations. It allows us to manipulate the equations in a way that sets up the elimination of a variable. By carefully choosing the multipliers, we can ensure that the coefficients align perfectly for cancellation. This step highlights the importance of precision and attention to detail in algebraic manipulations. A small error in the multiplication can throw off the entire solution. So, with our equations now prepped for elimination, let's move on to the next step and see how this strategic maneuver pays off!

Step 5: Eliminating 'z'

With our equations prepped and ready, let's eliminate 'z'! We'll do this by adding the modified second equation to the modified third equation. Remember, our goal is to make the 'z' terms disappear, leaving us with an equation that has only 'y' and 't'. Let's line up the equations and add them together:

 22y - 1.5z - 45t = 289.5  (Modified second equation)
 27.6y + 1.5z - 10.2t = -5.1  (Modified third equation)
--------------------------------------
 49.6y - 55.2t = 284.4        (Result of addition)

Great! The 'z' terms have canceled out, and we're left with a new equation that involves only 'y' and 't'. This is another significant step towards solving the system. We've managed to reduce the number of variables in another equation, making the system simpler to handle. This process showcases the power of strategic elimination in tackling complex problems. By carefully manipulating the equations, we can systematically reduce the number of unknowns and move closer to a solution. The resulting equation, 49.6y - 55.2t = 284.4, is a crucial piece of the puzzle. It establishes a relationship between 'y' and 't' that will help us solve for these variables. This step demonstrates that persistence and a systematic approach are key to success in solving systems of equations. So, with this new equation in hand, let's keep pushing forward and see how we can unravel the rest of the system!

Step 6: Back to Basics – Two Variables, Two Equations

Alright, we've made some serious headway! We've eliminated 'x' and 'z', and now we have a new equation, 49.6y - 55.2t = 284.4, that involves only 'y' and 't'. To solve for 'y' and 't', we need another equation that relates these two variables. Let's go back to our rewritten system from Step 3:

-0.2x + 0.3y + 0.4z - t = 4.4
 4.4y - 0.3z - 9t = 57.9
 9.2y + 0.5z - 3.4t = -1.7

We can use the second equation, 4.4y - 0.3z - 9t = 57.9, and the third equation, 9.2y + 0.5z - 3.4t = -1.7, to eliminate 'z' again, but this time in a slightly different way. This will give us another equation in terms of 'y' and 't'. We already did this in steps 4 and 5, but it's worth highlighting that we're essentially repeating a successful strategy to further simplify the system.

This step emphasizes the iterative nature of solving systems of equations. Often, we need to revisit previous steps and apply similar techniques to different parts of the system. It's like solving a maze – sometimes you need to backtrack and try a different path. By recognizing that we can reuse our elimination strategy, we're demonstrating a key problem-solving skill: adapting and applying known methods to new situations. The goal here is to create a manageable subsystem of equations that we can solve directly. So, with this in mind, let's proceed with our plan to eliminate 'z' and obtain another equation in 'y' and 't'!

Step 7: Creating a 2x2 System

As we discussed, we need to create a system of two equations with two variables ('y' and 't') to solve for those variables directly. We already have one such equation from Step 5: 49.6y - 55.2t = 284.4. To get the second equation, we'll revisit the elimination of 'z' from the second and third equations of our rewritten system (from Step 3). We essentially already did this in Steps 4 and 5, so let's recap the process and results:

We multiplied the second equation (4.4y - 0.3z - 9t = 57.9) by 5 to get:

22y - 1.5z - 45t = 289.5

We multiplied the third equation (9.2y + 0.5z - 3.4t = -1.7) by 3 to get:

27.6y + 1.5z - 10.2t = -5.1

Then, we added these two equations together, which eliminated 'z' and gave us:

49.6y - 55.2t = 284.4

This is the first equation in our 2x2 system. Now, let's use the equations before eliminating 'z' to create the second equation. We'll use the equations:

4.4y - 0.3z - 9t = 57.9
9.2y + 0.5z - 3.4t = -1.7

To eliminate 'z', we can multiply the first equation by 5 and the second equation by 3 (as we did in Step 4), but this time, we'll focus on setting up a different elimination. Let's multiply the first equation by 0.5 and the second equation by 0.3. This will give us opposite 'z' coefficients:

Multiply the first equation by 0.5:

0.  5 * (4.4y - 0.3z - 9t) = 0.5 * 57.9
2.  2y - 0.15z - 4.5t = 28.95

Multiply the second equation by 0.3:

3.  3 * (9.2y + 0.5z - 3.4t) = 0.3 * (-1.7)
4.  76y + 0.15z - 1.02t = -0.51

Now, we have two new equations with opposite 'z' coefficients. This strategic choice of multipliers highlights the flexibility we have in solving systems of equations. We can choose different paths to elimination depending on what seems most efficient. By focusing on decimal multipliers, we're setting up a slightly different elimination process, which will lead us to a second equation in 'y' and 't'. So, with these equations ready, let's move on to the next step and actually eliminate 'z' to complete our 2x2 system!

Step 8: Final Elimination for 2x2 System

Okay, we've got our equations prepped with those nice, opposite 'z' coefficients. Now comes the satisfying part: eliminating 'z' and getting our second equation in 'y' and 't'. Let's add the two equations we created in Step 7:

 2.2y - 0.15z - 4.5t = 28.95
 2.76y + 0.15z - 1.02t = -0.51
----------------------------------
 5.96y - 5.52t = 28.44

Boom! The 'z' terms are gone, and we're left with another equation involving only 'y' and 't'. This is exactly what we wanted. We now have our second equation for our 2x2 system. This step is a perfect example of how a well-planned strategy can lead to a clean and efficient solution. By carefully choosing our multipliers and setting up the elimination, we were able to isolate the variables we needed. The resulting equation, 10.96y - 5.52t = 28.44, is a crucial component of our simplified system. It provides a new relationship between 'y' and 't' that, when combined with our previous equation, will allow us to solve for these variables. This step underscores the importance of methodical execution in algebraic manipulations. Every addition and subtraction must be performed with precision to ensure the accuracy of the result. So, with our second equation in hand, let's assemble our 2x2 system and prepare to solve for 'y' and 't'!

Step 9: The 2x2 System Unveiled

We've worked hard to get here, and now it's time to see the fruits of our labor! We've successfully created a system of two equations with two variables ('y' and 't'). Let's write it out clearly:

49.6y - 55.2t = 284.4
10.96y - 5.52t = 28.44

Isn't that beautiful? Compared to our original system, this looks much more manageable. We've taken a complex problem and broken it down into a simpler form. This is a key principle in problem-solving: divide and conquer. By systematically eliminating variables, we've reduced the complexity of the system until we were left with a 2x2 system that we can solve using standard techniques. This step is a moment of triumph in our mathematical journey. It's a chance to appreciate the power of algebraic manipulation and the effectiveness of our strategic approach. The 2x2 system is a gateway to the final solution. It represents a significant milestone in our quest to unravel the unknowns. So, let's take a moment to admire our handiwork and then dive into the next step: solving this simplified system!

Step 10: Solving the 2x2 System

Now that we have our 2x2 system, we can use a variety of methods to solve for 'y' and 't'. Let's use the elimination method again, as it's been working well for us. Looking at our system:

49.  6y - 55.2t = 284.4
50.  96y - 5.52t = 28.44

Notice that the coefficient of 't' in the second equation is exactly one-tenth of the coefficient of 't' in the first equation. This gives us a great opportunity for elimination. We can multiply the second equation by -10, which will make the 't' coefficients opposites:

Multiply the second equation by -10:

-10 * (10.96y - 5.52t) = -10 * 28.44
-109.6y + 55.2t = -284.4

Now, we have a modified second equation that's perfectly set up for elimination. This strategic choice of multiplier highlights the importance of observation and pattern recognition in problem-solving. By noticing the relationship between the 't' coefficients, we were able to choose a multiplier that made the elimination process straightforward. Let's add this modified equation to the first equation to eliminate 't':

49.  6y - 55.2t = 284.4
-109.  6y + 55.2t = -284.4
-------------------------
-60y = 0

Wow! That's a clean elimination. The 't' terms canceled out perfectly, and we're left with a simple equation in 'y'. This is a testament to the power of strategic elimination and the beauty of mathematical precision. The resulting equation, -60y = 0, is a major breakthrough. It tells us directly that the value of 'y' is 0. This is a significant step towards solving the entire system. So, with 'y' in hand, let's move on to the next phase and use this information to find the value of 't'!

Step 11: Finding 't'

We've struck gold! We found that y = 0. Now, let's use this information to find the value of 't'. We can substitute y = 0 into either of the equations in our 2x2 system. Let's use the second equation, as it looks a bit simpler:

10.  96y - 5.52t = 28.44

Substitute y = 0:

10.  96 * 0 - 5.52t = 28.44
-5.52t = 28.44

Now, we have a simple equation with just one variable, 't'. Let's solve for 't':

t = 28.44 / -5.52
t = -5.152173913 (approximately)

Okay, we've got 't'! This step highlights the power of back-substitution in solving systems of equations. Once we find the value of one variable, we can plug it back into the equations to find the values of the other variables. The resulting value of 't', approximately -5.15, is another key piece of the puzzle. We're now halfway there! We've found the values of 'y' and 't', and we're ready to tackle the remaining variables. So, with these values in hand, let's move on to the next phase and see how we can find 'x' and 'z'!

Step 12: Back-Substituting for 'z'

Alright, we know y = 0 and t ≈ -5.15. Let's use these values to find 'z'. We can go back to one of the equations in our rewritten system (from Step 3) that involves 'y', 'z', and 't'. Let's use the third equation:

9.  2y + 0.5z - 3.4t = -1.7

Substitute y = 0 and t ≈ -5.15:

9.  2 * 0 + 0.5z - 3.4 * (-5.15) = -1.7
10.  5z + 17.51 = -1.7

Now, we have a simple equation with just one variable, 'z'. Let's solve for 'z':

11.  5z = -1.7 - 17.51
12.  5z = -19.21
z = -19.21 / 0.5
z = -38.42

We've found 'z'! This step showcases the effectiveness of back-substitution in unraveling the unknowns in a system of equations. By plugging in the values we've already found, we can systematically reduce the complexity of the remaining equations. The resulting value of 'z', approximately -38.42, is another crucial piece of our solution. We're getting closer and closer to completing the puzzle! So, with 'y', 't', and 'z' in hand, let's move on to the final variable and see how we can find 'x'!

Step 13: The Final Piece – Solving for 'x'

We're in the home stretch! We know y = 0, t ≈ -5.15, and z ≈ -38.42. Now, we just need to find 'x'. Let's go back to the original first equation, as it involves 'x':

-0.2x + 0.3y + 0.4z - t = 4.4

Substitute the values of y, z, and t:

-0.2x + 0.3 * 0 + 0.4 * (-38.42) - (-5.15) = 4.4
-0.2x - 15.368 + 5.15 = 4.4

Now, we have a simple equation with just one variable, 'x'. Let's solve for 'x':

-0.2x = 4.4 + 15.368 - 5.15
-0.2x = 14.618
x = 14.618 / -0.2
x = -73.09

We did it! We found 'x'! This final step is a testament to our perseverance and strategic approach. By systematically eliminating variables and back-substituting, we were able to unravel the entire system of equations. The resulting value of 'x', approximately -73.09, completes our solution. We've conquered the challenge! So, let's take a moment to celebrate our success and then move on to the final step: verifying our solution!

Step 14: Verification – Did We Crack the Code?

We've arrived at a potential solution: x ≈ -73.09, y = 0, z ≈ -38.42, and t ≈ -5.15. But before we declare victory, it's crucial to verify our solution. We need to make sure that these values satisfy all three original equations. This is a critical step in the problem-solving process. It's like double-checking your work before submitting a final project. Verification ensures that we haven't made any errors along the way and that our solution is accurate. Let's substitute our values into each of the original equations and see if they hold true.

Equation 1:

-0.2x + 0.3y + 0.4z - t = 4.4
-0.2 * (-73.09) + 0.3 * 0 + 0.4 * (-38.42) - (-5.15) = 4.4
14.  618 - 15.368 + 5.15 ≈ 4.4
4.  4 ≈ 4.4 (Correct!)

Equation 2:

51.  2x + 1.1y - 4.7z + 2t = 9.5
52.  2 * (-73.09) + 1.1 * 0 - 4.7 * (-38.42) + 2 * (-5.15) = 9.5
-160.798 + 180.574 - 10.3 ≈ 9.5
53.  466 ≈ 9.5 (Correct!)

Equation 3:

54.  2y + 0.5z - 3.4t = -1.7
55.  2 * 0 + 0.5 * (-38.42) - 3.4 * (-5.15) = -1.7
-19.21 + 17.51 ≈ -1.7
-1.7 ≈ -1.7 (Correct!)

Our solution checks out! All three equations are satisfied (with minor rounding errors). This is a moment of triumph! We've not only found a solution, but we've also verified its accuracy. This step underscores the importance of thoroughness in mathematical problem-solving. It's not enough to simply arrive at an answer; we must also ensure that the answer is correct. So, with our solution verified, let's move on to the final step: presenting our results!

The Solution: A Mathematical Masterpiece!

Guys, we did it! We successfully solved the system of equations! After a journey filled with strategic eliminations, back-substitutions, and careful calculations, we've arrived at the solution:

• x ≈ -73.09
• y = 0
• z ≈ -38.42
• t ≈ -5.15

This solution represents the unique set of values that satisfies all three equations in our original system. It's like finding the perfect combination lock code that opens a mathematical vault. We've not only found the code, but we've also demonstrated the process of finding it. This journey has been a testament to the power of algebraic manipulation and the effectiveness of a systematic approach. We've learned how to break down a complex problem into manageable steps, how to strategically eliminate variables, and how to back-substitute to find the unknowns. These skills are valuable not just in mathematics, but in any problem-solving situation. So, let's celebrate our success and remember the lessons we've learned along the way!

Solving systems of equations can seem daunting at first, but by breaking it down step by step, we've shown that it's totally achievable. Keep practicing, and you'll be a master in no time! Stay tuned for more math adventures, and remember, every problem is just a puzzle waiting to be solved. Peace out!