Solving Systems Of Equations A Step By Step Guide
Hey guys! Ever feel like you're staring at a jumbled mess of equations and just want to throw your hands up in the air? Well, fear not! We're going to break down how to solve a system of equations step-by-step, making it super easy to understand. Today, we're tackling this system:
-3x + 6y - 2z = 5
7x + 2y + 9z = 19
-x + y - z = 0
Don't worry, it looks intimidating, but we'll conquer it together! Let's dive in!
Why Solving Systems of Equations Matters
Before we jump into the nitty-gritty, let's quickly chat about why solving systems of equations is actually useful. You might be thinking, "When am I ever going to use this in real life?" But trust me, this skill pops up in all sorts of places! Think about scenarios where you have multiple unknowns and multiple pieces of information relating them. This could be anything from figuring out the right mix of ingredients for a recipe to optimizing business decisions or even understanding complex scientific models. Mastering this skill opens doors to problem-solving in various fields.
Real-World Applications
Imagine you're trying to figure out the cost of different items when you only have information about the total cost of combinations of those items. Or perhaps you're balancing chemical equations, where you need to find the coefficients that satisfy multiple constraints. Systems of equations are the key! They provide a framework for representing and solving these kinds of problems, making them an indispensable tool in mathematics, science, engineering, economics, and many other disciplines. So, understanding how to solve them isn't just about acing your math test; it's about building a powerful skill for life.
Cracking the Code: Our Strategy
Okay, so how do we actually solve this beast of a system? There are a few methods we could use, like substitution or elimination. Today, we're going to focus on the elimination method, which is often a clean and efficient way to go. The basic idea behind elimination is to strategically add or subtract multiples of equations to eliminate variables, one at a time, until we're left with a single equation in a single variable. Then, we can solve for that variable and work our way back to find the others. Sounds like a plan, right?
Setting the Stage: Labeling Our Equations
To keep things organized (and trust me, organization is KEY when solving systems!), let's label our equations:
Equation 1: -3x + 6y - 2z = 5 Equation 2: 7x + 2y + 9z = 19 Equation 3: -x + y - z = 0
Now we have a roadmap! Labeling equations is a small step that makes a big difference in preventing confusion and errors. It's like giving each equation a name so we can easily refer to it later. With our equations labeled, we're ready to start the elimination process.
Step-by-Step: Eliminating 'x'
Our first goal is to eliminate one of the variables. Looking at our equations, 'x' seems like a good candidate. We can use Equation 3 (which has a simple '-x') to eliminate 'x' from Equations 1 and 2. Here's how:
Eliminating 'x' from Equation 1
To eliminate 'x' from Equation 1, we'll multiply Equation 3 by -3 and add it to Equation 1:
Equation 3 * (-3): 3x - 3y + 3z = 0
Now, add this modified equation to Equation 1:
(-3x + 6y - 2z) + (3x - 3y + 3z) = 5 + 0
This simplifies to:
3y + z = 5
Let's call this new equation Equation 4.
Eliminating 'x' from Equation 2
Next, we'll eliminate 'x' from Equation 2. This time, we'll multiply Equation 3 by 7 and add it to Equation 2:
Equation 3 * (7): -7x + 7y - 7z = 0
Now, add this modified equation to Equation 2:
(7x + 2y + 9z) + (-7x + 7y - 7z) = 19 + 0
This simplifies to:
9y + 2z = 19
We'll call this Equation 5.
The Result: A Simpler System
Awesome! We've successfully eliminated 'x' from Equations 1 and 2. Now we have a new, simpler system of equations:
Equation 4: 3y + z = 5 Equation 5: 9y + 2z = 19
Notice that these equations only involve 'y' and 'z'. We're getting closer! Eliminating a variable is a crucial step in solving systems, and we've just nailed it. Now, let's move on to the next variable.
Taming 'y': Eliminating Another Variable
Now that we have a system with just two variables, 'y' and 'z', we can eliminate one of them to solve for the other. Let's eliminate 'y'. To do this, we'll multiply Equation 4 by -3 and add it to Equation 5:
Equation 4 * (-3): -9y - 3z = -15
Add this to Equation 5:
(9y + 2z) + (-9y - 3z) = 19 + (-15)
This simplifies to:
-z = 4
So, z = -4. Woohoo! We've found the value of 'z'! Finding one variable is a major breakthrough, as it allows us to back-substitute and find the others. With 'z' in hand, let's move on to the next step.
Back-Substitution Magic: Finding 'y' and 'x'
Now that we know z = -4, we can use back-substitution to find the values of 'y' and 'x'. This is where the magic happens! We'll start by plugging the value of 'z' into one of our equations involving 'y' and 'z' (Equation 4 or 5) to solve for 'y'. Then, we'll plug the values of 'y' and 'z' into one of the original equations to solve for 'x'.
Solving for 'y'
Let's use Equation 4: 3y + z = 5
Substitute z = -4:
3y + (-4) = 5
3y = 9
y = 3
Great! We've found that y = 3. Back-substitution is a powerful technique that allows us to leverage the values we've already found to uncover the remaining unknowns. Now, let's use the values of 'y' and 'z' to solve for 'x'.
Solving for 'x'
We can use any of the original equations to solve for 'x'. Let's use Equation 3: -x + y - z = 0
Substitute y = 3 and z = -4:
-x + 3 - (-4) = 0
-x + 7 = 0
-x = -7
x = 7
Awesome! We've found that x = 7. We've successfully navigated the back-substitution process and uncovered the value of 'x'. Now, let's celebrate our solution!
The Grand Finale: Our Solution
We've done it! We've solved the system of equations! Our solution is:
x = 7 y = 3 z = -4
We can write this as an ordered triple: (7, 3, -4). This means that the values x = 7, y = 3, and z = -4 satisfy all three equations in the system. Finding the solution is the ultimate goal, and we've achieved it through careful application of the elimination method and back-substitution.
Double-Checking Our Work
Before we declare victory, it's always a good idea to double-check our solution. We can do this by plugging the values of x, y, and z back into the original equations to make sure they hold true. Let's try it:
Equation 1: -3x + 6y - 2z = 5
-3(7) + 6(3) - 2(-4) = -21 + 18 + 8 = 5 (Correct!)
Equation 2: 7x + 2y + 9z = 19
7(7) + 2(3) + 9(-4) = 49 + 6 - 36 = 19 (Correct!)
Equation 3: -x + y - z = 0
-7 + 3 - (-4) = -7 + 3 + 4 = 0 (Correct!)
Our solution checks out! Verifying our solution is a crucial step in the problem-solving process. It gives us confidence that we've arrived at the correct answer and haven't made any errors along the way.
Congrats, Mathletes!
And there you have it! We've successfully solved a system of three equations with three variables. Give yourselves a pat on the back! Remember, the key to mastering these problems is to stay organized, take it step-by-step, and don't be afraid to ask for help if you get stuck. You've got this! Keep practicing, and you'll be solving systems of equations like a pro in no time!
