Solving A System Of 3 Equations: A Step-by-Step Guide
Hey guys! Ever stumbled upon a system of equations that looks like a tangled mess? Don't worry, it happens to the best of us! Today, we're going to break down a specific system of three equations and show you exactly how to solve it. We'll go through each step in detail, so you can confidently tackle similar problems in the future. Let's dive in!
The Challenge: Our System of Equations
We are faced with the following system of equations:
It might look intimidating at first, but trust me, it's manageable. Our goal is to find the values of x, y, and z that satisfy all three equations simultaneously. There are several methods to achieve this, but we'll focus on a combination of substitution and elimination – powerful techniques in the world of algebra.
Understanding the Equations
Before we jump into solving, let's take a closer look at what we're dealing with. Each equation represents a plane in 3D space. The solution to the system, if one exists, is the point where all three planes intersect. This point has three coordinates: the values of x, y, and z that make all equations true.
Why This Matters
Solving systems of equations isn't just a math exercise; it's a fundamental skill with applications across various fields. From engineering and physics to economics and computer science, these systems are used to model and solve real-world problems. Mastering this skill opens doors to a deeper understanding of these fields.
Step-by-Step Solution: Cracking the Code
Now, let's get our hands dirty and solve this system! We'll use a combination of elimination and substitution to systematically reduce the complexity of the system until we can isolate the value of each variable.
Step 1: Elimination – Targeting a Variable
The first equation that catches my eye is the second one: x + 2z = 4. It's simpler than the others, and we can easily isolate x. Let's rearrange it:
x = 4 - 2z
This is a crucial first step! We've expressed x in terms of z, which means we can substitute this expression into the other equations to eliminate x from them. This simplifies the system and brings us closer to a solution.
Step 2: Substitution – Replacing and Simplifying
Now we substitute the expression for x (4 - 2z) into the first and third equations:
- Equation 1: 5(4 - 2z) + 2y + z = 4
- Equation 3: 2(4 - 2z) + y - z = -1
Let's simplify these equations by expanding and combining like terms:
- Simplified Equation 1: 20 - 10z + 2y + z = 4 => 2y - 9z = -16
- Simplified Equation 3: 8 - 4z + y - z = -1 => y - 5z = -9
Look at what we've done! We've transformed our original system into a new system with only two equations and two variables (y and z). This is a huge step forward. We're now dealing with a much more manageable problem.
Step 3: Elimination Again – Knocking Out Another Variable
We now have a system of two equations:
Let's eliminate y. To do this, we can multiply the second equation by -2:
-2(y - 5z) = -2(-9) => -2y + 10z = 18
Now we add this modified equation to the first equation:
(2y - 9z) + (-2y + 10z) = -16 + 18 => z = 2
Boom! We've found the value of z. This is a major breakthrough. With one variable solved, the rest will fall into place.
Step 4: Back-Substitution – Unraveling the Mystery
Now that we know z = 2, we can use back-substitution to find the values of x and y. Let's start with the equation x = 4 - 2z:
x = 4 - 2(2) = 4 - 4 = 0
So, x = 0. We're on a roll!
Next, let's substitute z = 2 into the equation y - 5z = -9:
y - 5(2) = -9 => y - 10 = -9 => y = 1
And there we have it! y = 1.
Step 5: The Solution – Victory is Ours!
We've found the values of all three variables:
- x = 0
- y = 1
- z = 2
Therefore, the solution to the system of equations is x = 0, y = 1, z = 2. This corresponds to option A in the original problem.
Answer verification
Let's verify if the solution is correct. We will plug in the values of x, y and z in the original equations.
- Equation 1: 5(0) + 2(1) + 2 = 0 + 2 + 2 = 4 (Correct)
- Equation 2: 0 + 2(2) = 0 + 4 = 4 (Correct)
- Equation 3: 2(0) + 1 - 2 = 0 + 1 - 2 = -1 (Correct)
Alternative solution
We can solve the problem by plugging in the values of the options in the equations, and verifying if the equations are true.
Option A
- Equation 1: 5(0) + 2(1) + 2 = 0 + 2 + 2 = 4 (Correct)
- Equation 2: 0 + 2(2) = 0 + 4 = 4 (Correct)
- Equation 3: 2(0) + 1 - 2 = 0 + 1 - 2 = -1 (Correct)
Option A is the correct answer.
Key Takeaways: Mastering the Art of Solving Systems
Solving systems of equations can seem tricky, but with a systematic approach, you can conquer them! Here are some key takeaways from our journey:
- Elimination is your friend: Look for opportunities to eliminate variables by adding or subtracting multiples of equations.
- Substitution is powerful: Expressing one variable in terms of others allows you to simplify the system.
- Back-substitution is the key to unraveling: Once you find the value of one variable, plug it back into the equations to find the others.
- Verify your solution: Always double-check your answer by plugging the values back into the original equations.
Practice Makes Perfect
The best way to master solving systems of equations is to practice. Work through different examples, experiment with different methods, and don't be afraid to make mistakes – that's how we learn!
Where to Go Next
If you're looking to further hone your skills, consider exploring these topics:
- Matrices and determinants: These tools provide a powerful way to represent and solve systems of equations.
- Gaussian elimination: A systematic algorithm for solving linear systems.
- Applications of systems of equations: Explore how these systems are used in real-world scenarios.
Conclusion: You've Got This!
So, guys, we've successfully tackled a system of three equations! Remember, the key is to break down the problem into smaller, manageable steps. With practice and the right techniques, you'll be solving these systems like a pro in no time. Keep exploring, keep learning, and most importantly, keep having fun with math!