Solving Systems Of Equations A Step By Step Guide

Hey guys! Ever found yourselves staring blankly at a system of equations, feeling like you're trying to decipher ancient hieroglyphs? Don't worry, you're not alone! Systems of equations can seem daunting, but with the right approach, they're totally conquerable. In this article, we're going to break down a specific system of equations step-by-step and explore the solution. So, grab your thinking caps, and let's dive in!

The System at Hand

Our mission, should we choose to accept it, is to solve the following system of equations:

3x + 2y + z = 20
x - 4y - z = -10
2x + y + 2z = 15

This looks like a puzzle, right? We have three equations and three unknowns (x, y, and z). Our goal is to find the values of these variables that satisfy all three equations simultaneously. There are several methods to tackle this, but we'll focus on the elimination method, which is a classic and effective technique.

The Elimination Method Explained

The elimination method is all about strategically adding or subtracting multiples of equations to eliminate one variable at a time. This simplifies the system, making it easier to solve. The core idea is to manipulate the equations so that the coefficients of one variable are opposites. When we add the equations, that variable cancels out, leaving us with a simpler equation in fewer variables.

Think of it like this: imagine you have a bunch of building blocks, and you want to combine them to make a simpler structure. The elimination method is like carefully choosing which blocks to combine so that some parts cancel each other out, leaving you with the essential elements. Let's see how this works in practice!

Step-by-Step Solution

Let's get our hands dirty and solve this system. We'll label the equations for easy reference:

Equation 1: 3x + 2y + z = 20

Equation 2: x - 4y - z = -10

Equation 3: 2x + y + 2z = 15

Eliminating z First

Notice that Equation 1 and Equation 2 have z and -z terms. This is perfect for elimination! If we add these two equations together, the z terms will cancel out:

(3x + 2y + z) + (x - 4y - z) = 20 + (-10)

Combining like terms, we get:

4x - 2y = 10

Let's call this new equation Equation 4. Now we have an equation with only x and y.

To eliminate z again, we can work with Equation 1 and Equation 3. To make the z coefficients opposites, we'll multiply Equation 1 by -2:

-2 * (3x + 2y + z) = -2 * 20

This gives us:

-6x - 4y - 2z = -40

Now, let's add this modified equation to Equation 3:

(-6x - 4y - 2z) + (2x + y + 2z) = -40 + 15

Combining like terms, we get:

-4x - 3y = -25

Let's call this Equation 5. Now we have two equations (Equation 4 and Equation 5) with two variables (x and y):

Equation 4: 4x - 2y = 10

Equation 5: -4x - 3y = -25

Solving for x and y

Guess what? We can eliminate x easily now! Notice that the x coefficients in Equation 4 and Equation 5 are 4 and -4. If we add these equations together, the x terms will vanish:

(4x - 2y) + (-4x - 3y) = 10 + (-25)

Combining like terms, we have:

-5y = -15

Dividing both sides by -5, we find:

y = 3

Awesome! We've found the value of y. Now we can substitute this value back into either Equation 4 or Equation 5 to solve for x. Let's use Equation 4:

4x - 2(3) = 10

Simplifying:

4x - 6 = 10

Adding 6 to both sides:

4x = 16

Dividing both sides by 4:

x = 4

Fantastic! We've found both x and y. Now we just need to find z.

Solving for z

We can substitute the values of x and y into any of the original equations to solve for z. Let's use Equation 1:

3(4) + 2(3) + z = 20

Simplifying:

12 + 6 + z = 20

18 + z = 20

Subtracting 18 from both sides:

z = 2

Eureka! We've found z. So, the solution to the system of equations is x = 4, y = 3, and z = 2.

The Grand Finale The Solution

Therefore, the solution to the system of equations is (4, 3, 2). This corresponds to option C in the given choices. We did it! We successfully navigated the maze of equations and emerged victorious.

Key Takeaways and Other Methods

Alternative Methods for Solving Systems of Equations

While we've focused on the elimination method, it's worth mentioning that there are other powerful techniques for solving systems of equations. Here's a quick overview:

1. Substitution Method: In this method, you solve one equation for one variable and then substitute that expression into the other equations. This reduces the number of variables in the remaining equations.

2. Matrix Methods: For larger systems of equations, matrix methods like Gaussian elimination or using the inverse of a matrix can be very efficient. These methods involve representing the system as a matrix and then performing operations to solve for the variables.

3. Graphical Method: For systems with two variables, you can graph the equations and find the point(s) of intersection. This method provides a visual representation of the solution.

Importance of Checking Your Solution

It's always a good practice to check your solution by substituting the values back into the original equations. This ensures that the solution satisfies all the equations and that you haven't made any errors along the way. For example, plugging x = 4, y = 3, and z = 2 into the original equations:

Equation 1: 3(4) + 2(3) + 2 = 12 + 6 + 2 = 20 (Correct!)

Equation 2: 4 - 4(3) - 2 = 4 - 12 - 2 = -10 (Correct!)

Equation 3: 2(4) + 3 + 2(2) = 8 + 3 + 4 = 15 (Correct!)

Since the solution satisfies all three equations, we can be confident that it's correct.

Real-World Applications

Systems of equations might seem like an abstract mathematical concept, but they have tons of real-world applications. Here are just a few examples:

• Engineering: Systems of equations are used to analyze circuits, design structures, and model fluid flow.

• Economics: Economists use systems of equations to model supply and demand, analyze market equilibrium, and forecast economic trends.

• Computer Graphics: Systems of equations are used to create 3D models, render images, and simulate physics.

• Chemistry: Chemists use systems of equations to balance chemical reactions and calculate concentrations.

• Navigation: GPS systems use systems of equations to determine your location based on signals from satellites.

Tips for Mastering Systems of Equations

1. Practice, practice, practice! The more you work through different types of systems, the more comfortable you'll become with the techniques.

2. Stay organized. Keep your work neat and clearly label your equations. This will help you avoid mistakes and make it easier to track your progress.

3. Check your solutions. Always substitute your solutions back into the original equations to verify that they're correct.

4. Don't be afraid to experiment. Try different methods and see which ones work best for you. There's no one-size-fits-all approach.

5. Seek help when needed. If you're stuck, don't hesitate to ask your teacher, classmates, or an online forum for help.

Conclusion: You've Got This!

So, there you have it! We've successfully solved a system of equations using the elimination method. Remember, the key is to break down the problem into smaller steps, stay organized, and practice consistently. With a little bit of effort, you'll be solving systems of equations like a pro in no time!

Solving systems of equations might seem challenging at first, but with a systematic approach and a bit of practice, you can conquer them. The elimination method is a powerful tool, and understanding its principles can help you tackle a wide range of problems. Remember to always check your solutions and explore different methods to find what works best for you. Whether it's for math class or real-world applications, mastering systems of equations is a valuable skill. So keep practicing, and don't be afraid to ask for help when you need it. You've got this!

Now go forth and conquer those equations!