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

Hey math enthusiasts! Today, we're diving into the fascinating world of systems of equations. Specifically, we're going to break down how to solve the following system. If you've ever felt lost in a maze of variables, don't worry; we're here to guide you through it. Let's make this crystal clear so you can tackle any similar problem with confidence! Get ready to sharpen your pencils, because we're about to embark on a mathematical journey that will not only demystify this particular system but also equip you with the skills to conquer any equation-solving challenge that comes your way. So, buckle up and let’s get started!

The Challenge: Our System of Equations

Before we get started, let's lay out the equations we're tackling:

$\begin{aligned} 3 x-y+z & =-3 \\ -2 x+y-2 z & =8 \\ -4 x+3 y-z & =6 \end{aligned}$

Systems of equations might seem intimidating at first glance, but they're really just a set of equations with multiple variables that we need to solve simultaneously. The goal? Find the values for x, y, and z that satisfy all three equations at once. There are several methods we can use, but we'll focus on a combination of elimination and substitution. This approach is super versatile and will help you solve all sorts of systems, even the tricky ones. Think of it like a puzzle – each equation is a clue, and we're piecing them together to find the solution. Stick with us, and you'll see how fun and rewarding this process can be!

Step 1: Elimination Time!

Our first key step is to eliminate one of the variables. Let's aim for eliminating y because it looks like it has some easy coefficients to work with. Notice how the first and second equations have -y and +y, respectively? This is perfect for elimination! We'll add these two equations together. This is a common strategy in solving systems of equations, and it's all about finding the most efficient path to the solution. By eliminating one variable, we simplify the problem and make it easier to manage. So, let's roll up our sleeves and get ready to add those equations together, because we're one step closer to cracking this mathematical puzzle!

Combining Equations 1 and 2

Adding the first and second equations, we get:

(3x - y + z) + (-2x + y - 2z) = -3 + 8

Simplifying this, we have:

x - z = 5

We'll call this new equation Equation 4. This is a huge win! We've managed to eliminate y and create a simpler equation with just x and z. This is a classic move in the world of systems of equations, and it's all about strategic simplification. By reducing the number of variables in our equation, we're making the problem much more manageable. Think of it as clearing a path through the mathematical jungle – we're removing obstacles and making our journey to the solution smoother. So, let's keep this momentum going and see what other variables we can knock out!

Step 2: Eliminating y Again

Now, let's eliminate y again, but this time using a different pair of equations. We'll use the first and third equations. The first equation has -y, and the third has 3y, so we'll multiply the first equation by 3 to match the y coefficients. This might seem like a bit of a detour, but trust us, it's a necessary step to solve the puzzle. Remember, the goal is to isolate variables and make the equations simpler to handle. By strategically multiplying and combining equations, we're setting ourselves up for success. So, let's get those multipliers in place and continue our quest for the solution!

Adjusting Equation 1

Multiply Equation 1 by 3:

3 * (3x - y + z) = 3 * -3

This gives us:

9x - 3y + 3z = -9

Now, let's call this adjusted equation Equation 1'.

Combining Equations 1' and 3

Add Equation 1' to Equation 3:

(9x - 3y + 3z) + (-4x + 3y - z) = -9 + 6

Simplifying, we get:

5x + 2z = -3

Let's call this Equation 5. Awesome! We've eliminated y once more, and now we have another equation with just x and z. This is exactly what we wanted – a pair of equations that we can solve together to find the values of these two variables. Think of it like narrowing down the suspects in a mystery novel – we're getting closer and closer to the truth. So, let's keep up the good work and see how we can use these new equations to finally crack the case!

Step 3: Solving for x and z

We now have two equations with two variables:

Equation 4: x - z = 5

Equation 5: 5x + 2z = -3

Let's use these to solve for x and z. There are several ways to do this, but we'll use the elimination method again. We can multiply Equation 4 by 2 to make the z coefficients easier to work with. This is all about finding the most efficient way to untangle the equations and isolate the variables we're after. Think of it as navigating a maze – we're looking for the quickest path to the exit, and sometimes that means taking a slightly different route. So, let's get those multipliers ready and see how we can use them to simplify our equations and get closer to the solution!

Adjusting Equation 4

Multiply Equation 4 by 2:

2 * (x - z) = 2 * 5

This gives us:

2x - 2z = 10

Let's call this Equation 4'.

Combining Equations 4' and 5

Add Equation 4' to Equation 5:

(2x - 2z) + (5x + 2z) = 10 + (-3)

Simplifying, we have:

7x = 7

Dividing both sides by 7, we get:

x = 1

Yes! We've found the value of x! This is a major breakthrough, guys! It's like finding the first piece of a puzzle – now we can use it to piece together the rest of the solution. We've successfully navigated the tricky twists and turns of the equations, and we're one step closer to cracking the code. So, let's take a moment to celebrate this victory, and then dive back in to see how we can use this newfound knowledge to find the values of the other variables. The end is in sight, so let's keep up the great work!

Finding z

Now that we know x = 1, we can substitute this value into Equation 4:

1 - z = 5

Subtracting 1 from both sides:

-z = 4

Multiplying both sides by -1:

z = -4

Fantastic! We've nailed down the value of z as well! With x and z in our grasp, we're really starting to see the full picture. It's like watching a beautiful landscape emerge from the fog – the solution is becoming clearer and clearer. We've conquered the challenges of elimination and substitution, and now we're ready to move on to the final stage of our mathematical journey. So, let's take a deep breath and prepare to use our newfound knowledge to unlock the value of the last remaining variable. We're almost there, guys!

Step 4: Solving for y

With x = 1 and z = -4, we can substitute these values into any of the original equations to solve for y. Let's use Equation 1:

3x - y + z = -3

Substituting the values, we get:

3 * (1) - y + (-4) = -3

Simplifying:

3 - y - 4 = -3

-1 - y = -3

Adding 1 to both sides:

-y = -2

Multiplying both sides by -1:

y = 2

The Final Solution

We've done it! We've successfully solved the system of equations. The solution is:

• x = 1
• y = 2
• z = -4

So, the solution set is (1, 2, -4). Give yourselves a pat on the back, guys! We've navigated the twists and turns of the equations, conquered the challenges of elimination and substitution, and emerged victorious with the solution in hand. This is a testament to your perseverance and problem-solving skills. Remember, the journey might have seemed daunting at first, but with a little strategy and a lot of determination, you can conquer any mathematical challenge. So, keep practicing, keep exploring, and never stop believing in your ability to solve!