Solving Systems Of Equations A Step By Step Guide

Hey guys! Ever found yourselves staring blankly at a system of equations, wondering where to even begin? You're definitely not alone! Systems of equations can seem intimidating at first, but with the right approach, they become totally manageable. In this article, we're going to dive deep into how to solve systems of equations, using a real example to guide us. We'll break down each step, explain the logic behind it, and make sure you're equipped to tackle similar problems on your own. So, let's get started and turn those equation-solving frowns upside down!

Understanding Systems of Equations

Before we jump into solving, let's quickly recap what a system of equations actually is. Basically, it's a set of two or more equations that share the same variables. Our goal is to find the values for these variables that satisfy all equations simultaneously. Think of it as a puzzle where each equation gives you a clue, and you need to piece them together to find the solution. There are several methods to solve these systems, including substitution, elimination, and matrix methods. In our example, we'll focus on the elimination method, which is a powerful technique for systematically eliminating variables.

The elimination method is a strategic approach that involves manipulating the equations in the system to eliminate one variable at a time. This is achieved by multiplying one or more equations by constants so that when you add or subtract the equations, one of the variables cancels out. Once you've eliminated a variable, you're left with a simpler system that you can solve for the remaining variables. The key to success with the elimination method is careful organization and attention to detail. You need to keep track of your equations and the operations you perform on them. It might sound a bit complex now, but trust me, it becomes much clearer as we walk through our example step-by-step.

Now, let's talk about why understanding and solving systems of equations is super important. It's not just some abstract math concept! Systems of equations pop up all over the place in real-world applications. For example, they're used in economics to model supply and demand, in physics to analyze motion and forces, and in engineering to design structures and circuits. Even in everyday life, you might encounter situations where you need to solve a system of equations without even realizing it, like figuring out the best combination of ingredients for a recipe or planning a budget. So, mastering this skill can really open doors and help you in various fields.

The System at Hand

Okay, let's get to the heart of the matter. The system of equations we're going to solve is:

9x - y + z = 1
4x + 2y - 3z = 25
x - 3y + 2z = -18

We've got three equations with three unknowns: x, y, and z. Our mission, should we choose to accept it (and we do!), is to find the values for these variables that make all three equations true. This might seem like a daunting task, but don't worry! We'll break it down into manageable steps, using the elimination method to guide us. We'll systematically eliminate variables, one at a time, until we're left with a single equation that we can easily solve. Then, we'll work our way back, substituting the values we find to determine the other variables. It's like solving a puzzle, and the satisfaction of finding the solution is totally worth the effort.

Before we dive into the elimination process, it's a good idea to take a moment to look at the equations and think about our strategy. Which variables seem easiest to eliminate? Are there any equations that we can easily manipulate to make the elimination process smoother? These are the kinds of questions we want to be asking ourselves. For example, in this system, we might notice that the y terms have coefficients of -1, 2, and -3. This suggests that we might be able to eliminate y fairly easily by multiplying the first equation by 2 and adding it to the second equation, or by multiplying the first equation by -3 and adding it to the third equation. We'll see exactly how this works in the next section.

Step-by-Step Solution

Alright, let's roll up our sleeves and get into the nitty-gritty of solving this system. We'll tackle this systematically, so you can see each step clearly.

Step 1: Eliminating y from the first two equations

Our first move is to eliminate the variable y from the first two equations. To do this, we'll multiply the first equation by 2. This will give us a -2y term, which will nicely cancel out the +2y term in the second equation.

So, multiplying the first equation (9x - y + z = 1) by 2, we get:

18x - 2y + 2z = 2

Now, we'll add this modified equation to the second equation (4x + 2y - 3z = 25):

(18x - 2y + 2z) + (4x + 2y - 3z) = 2 + 25

Combining like terms, we get:

22x - z = 27

Let's call this new equation Equation (4). We've successfully eliminated y from the first two equations, and now we have a new equation relating x and z.

Step 2: Eliminating y from the first and third equations

Next up, we need to eliminate y from another pair of equations. Let's use the first and third equations this time. To eliminate y, we'll multiply the first equation (9x - y + z = 1) by -3. This will give us a +3y term, which will cancel out the -3y term in the third equation.

Multiplying the first equation by -3, we get:

-27x + 3y - 3z = -3

Now, we'll add this modified equation to the third equation (x - 3y + 2z = -18):

(-27x + 3y - 3z) + (x - 3y + 2z) = -3 + (-18)

Combining like terms, we get:

-26x - z = -21

Let's call this Equation (5). Now we have another equation relating x and z.

Step 3: Solving for x

Now we have two equations (Equation (4) and Equation (5)) with only two variables (x and z). This is a smaller system that we can solve more easily. Let's rewrite these equations:

22x - z = 27  (Equation 4)
-26x - z = -21 (Equation 5)

To eliminate z, we can subtract Equation (5) from Equation (4):

(22x - z) - (-26x - z) = 27 - (-21)

Simplifying, we get:

48x = 48

Dividing both sides by 48, we find:

x = 1

Yay! We've found the value of x.

Step 4: Solving for z

Now that we know x = 1, we can substitute this value into either Equation (4) or Equation (5) to solve for z. Let's use Equation (4):

22x - z = 27

Substituting x = 1, we get:

22(1) - z = 27

Simplifying, we have:

22 - z = 27

Subtracting 22 from both sides, we get:

-z = 5

Multiplying both sides by -1, we find:

z = -5

Awesome! We've found the value of z.

Step 5: Solving for y

We're almost there! Now that we know x = 1 and z = -5, we can substitute these values into any of the original equations to solve for y. Let's use the first equation (9x - y + z = 1):

9x - y + z = 1

Substituting x = 1 and z = -5, we get:

9(1) - y + (-5) = 1

Simplifying, we have:

9 - y - 5 = 1

4 - y = 1

Subtracting 4 from both sides, we get:

-y = -3

Multiplying both sides by -1, we find:

y = 3

We did it! We've found the value of y.

The Solution

So, after all that work, what's the solution to our system of equations? We found that:

• x = 1
• y = 3
• z = -5

This means that the ordered triple (1, 3, -5) is the solution to the system. This is the one and only set of values for x, y, and z that satisfies all three equations simultaneously.

Checking Our Work

But wait, we're not done yet! It's always a good idea to check our solution to make sure we haven't made any mistakes along the way. To do this, we'll substitute our values for x, y, and z back into the original equations and see if they hold true.

Let's start with the first equation:

9x - y + z = 1

Substituting x = 1, y = 3, and z = -5, we get:

9(1) - 3 + (-5) = 1

9 - 3 - 5 = 1

1 = 1

Great! The first equation checks out.

Now, let's check the second equation:

4x + 2y - 3z = 25

Substituting x = 1, y = 3, and z = -5, we get:

4(1) + 2(3) - 3(-5) = 25

4 + 6 + 15 = 25

25 = 25

Excellent! The second equation checks out too.

Finally, let's check the third equation:

x - 3y + 2z = -18

Substituting x = 1, y = 3, and z = -5, we get:

1 - 3(3) + 2(-5) = -18

1 - 9 - 10 = -18

-18 = -18

Fantastic! The third equation also checks out. Since our solution satisfies all three equations, we can be confident that it's correct.

Conclusion

And there you have it! We've successfully solved a system of three equations with three unknowns using the elimination method. We've broken down each step, explained the reasoning behind it, and even checked our work to make sure we got the right answer. Solving systems of equations might seem tricky at first, but with practice and a systematic approach, you can master this important skill. Remember, the key is to stay organized, pay attention to detail, and don't be afraid to ask for help if you get stuck. Now go out there and conquer those equations!

Key Takeaways:

• The elimination method is a powerful technique for solving systems of equations.
• Eliminate variables one at a time by multiplying and adding/subtracting equations.
• Check your solution by substituting the values back into the original equations.
• Practice makes perfect! The more you solve, the more confident you'll become.

Practice Problems

To really solidify your understanding, try solving these practice problems:

1. Solve the system:

   2x + y - z = 3
   x - y + 2z = 0
   3x + 2y + z = 5
   

2. Solve the system:

   x + 2y + 3z = 14
   2x - y + z = 3
   3x + y - 2z = 2
   

Good luck, and happy equation solving!