Solving Systems Of Equations A Step-by-Step Guide

by Axel Sørensen 50 views

Solving systems of equations might seem daunting at first, but don't worry, guys! With a systematic approach and a little practice, you'll be able to tackle these problems like a pro. This guide will walk you through a detailed solution to the following system of equations:

2x + 3y - 2z = -7
x - 2y + 4z = 15
2y + z = 1

We'll explore the steps involved in solving this system using the elimination method, a powerful technique for finding the values of x, y, and z that satisfy all three equations simultaneously.

Understanding Systems of Equations

Before we dive into the solution, let's take a moment to understand what a system of equations actually represents. A system of equations is simply a set of two or more equations that share the same variables. The goal is to find the values for these variables that make all the equations true at the same time. Think of it like finding the common ground where all the equations agree.

In our case, we have three equations with three variables (x, y, and z). This means we're looking for a specific set of values for x, y, and z that will make each of the three equations a true statement. There are several methods to solve these systems, but the elimination method is particularly useful when dealing with linear equations like the ones we have here. The elimination method focuses on strategically adding or subtracting multiples of equations to eliminate one variable at a time, making the system simpler to solve.

The beauty of systems of equations lies in their ability to model real-world situations. From calculating the optimal mix of ingredients in a recipe to determining the trajectory of a rocket, systems of equations are used extensively in various fields like engineering, economics, and computer science. Mastering the techniques to solve these systems opens up a world of possibilities for applying mathematics to practical problems. So, let's get started and unlock the secrets to solving these fascinating mathematical puzzles!

Step-by-Step Solution Using Elimination Method

The elimination method is a fantastic technique for solving systems of equations. The core idea is to manipulate the equations so that when you add or subtract them, one of the variables cancels out. This simplifies the system, allowing you to solve for the remaining variables. Let's break down the process step-by-step for our system:

1. Label the Equations

First, let's label our equations for easy reference:

  • Equation 1: 2x + 3y - 2z = -7
  • Equation 2: x - 2y + 4z = 15
  • Equation 3: 2y + z = 1

This might seem like a small step, but it's crucial for keeping track of your work and avoiding confusion as we move through the elimination process. Clear labeling is a cornerstone of organized problem-solving, especially when dealing with multiple equations and variables.

2. Eliminate 'x' from Equations 1 and 2

Our goal here is to get rid of the 'x' variable in one of the equations. To do this, we'll multiply Equation 2 by -2. This will give us a '-2x' term, which will cancel out the '2x' term in Equation 1 when we add the equations together.

  • Multiply Equation 2 by -2: -2(x - 2y + 4z) = -2(15) => -2x + 4y - 8z = -30

Now, let's add this modified version of Equation 2 to Equation 1:

  2x + 3y - 2z = -7
+ (-2x + 4y - 8z = -30)
---------------------
  0x + 7y - 10z = -37

We've successfully eliminated 'x'! Let's call this new equation Equation 4:

  • Equation 4: 7y - 10z = -37

By strategically multiplying and adding equations, we've reduced our system to one with fewer variables, bringing us closer to the solution. This is the essence of the elimination method: simplify, simplify, simplify!

3. Eliminate 'y' from Equations 3 and 4

Now we have two equations (Equation 3 and Equation 4) that only contain 'y' and 'z'. Our next step is to eliminate another variable, and 'y' seems like a good candidate. To do this, we need to manipulate the equations so that the 'y' coefficients are opposites. Let's multiply Equation 3 by -7/2:

  • Multiply Equation 3 by -7/2: (-7/2)(2y + z) = (-7/2)(1) => -7y - (7/2)z = -7/2

Now, we can add this modified version of Equation 3 to Equation 4:

  7y - 10z = -37
+ (-7y - (7/2)z = -7/2)
----------------------
  0y - (27/2)z = -81/2

We've eliminated 'y'! Let's simplify the resulting equation:

  • -(27/2)z = -81/2

To solve for 'z', we can multiply both sides by -2/27:

  • z = (-81/2) * (-2/27) = 3

Fantastic! We've found the value of 'z'. By carefully choosing our multipliers and adding the equations, we've successfully isolated 'z' and solved for its value. This victory brings us closer to unraveling the entire system of equations.

4. Substitute 'z' into Equation 3 to find 'y'

Now that we know z = 3, we can substitute this value back into Equation 3 (2y + z = 1) to solve for 'y':

  • 2y + 3 = 1
  • 2y = -2
  • y = -1

Excellent! We've found the value of 'y'. Substitution is a powerful technique in solving systems of equations. Once you find the value of one variable, you can plug it back into an equation containing that variable to solve for another. This cascading effect allows you to systematically unravel the solution.

5. Substitute 'y' and 'z' into Equation 2 to find 'x'

Finally, we can substitute the values of y = -1 and z = 3 into Equation 2 (x - 2y + 4z = 15) to solve for 'x':

  • x - 2(-1) + 4(3) = 15
  • x + 2 + 12 = 15
  • x + 14 = 15
  • x = 1

We've done it! We've found the value of 'x'. By systematically substituting the values we found for 'y' and 'z', we've successfully isolated 'x' and completed the solution.

The Solution

Therefore, the solution to the system of equations is:

  • x = 1
  • y = -1
  • z = 3

We can write this as an ordered triple: (1, -1, 3).

Congratulations! You've successfully navigated the elimination method and solved a system of three equations with three variables. Remember, practice makes perfect, so keep working on these types of problems to build your skills and confidence.

Verification: The Key to Accuracy

Before we celebrate our victory, it's crucial to verify our solution. This is a simple but essential step to ensure that our calculated values are correct and satisfy all the equations in the system. Verification involves plugging the values we found (x = 1, y = -1, z = 3) back into the original equations and checking if the equations hold true.

1. Substitute into Equation 1: 2x + 3y - 2z = -7

  • 2(1) + 3(-1) - 2(3) = -7
  • 2 - 3 - 6 = -7
  • -7 = -7 (This is true!)

2. Substitute into Equation 2: x - 2y + 4z = 15

  • 1 - 2(-1) + 4(3) = 15
  • 1 + 2 + 12 = 15
  • 15 = 15 (This is true!)

3. Substitute into Equation 3: 2y + z = 1

  • 2(-1) + 3 = 1
  • -2 + 3 = 1
  • 1 = 1 (This is true!)

Our solution (x = 1, y = -1, z = 3) satisfies all three equations! This confirms that our solution is correct. Verification is not just a formality; it's a powerful tool for catching errors and building confidence in your problem-solving abilities. Always take the time to verify your answers, especially in high-stakes situations like exams.

Alternative Methods for Solving Systems of Equations

While the elimination method is a powerful technique, it's not the only way to solve systems of equations. There are other methods available, each with its own strengths and weaknesses. Understanding these different approaches can broaden your problem-solving toolkit and allow you to choose the most efficient method for a given system.

1. Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into another equation. This eliminates one variable, allowing you to solve for the remaining variable. You can then substitute the value you found back into one of the original equations to solve for the other variable. This method is particularly useful when one of the equations is already solved for a variable or can be easily solved.

2. Matrix Methods

For larger systems of equations, matrix methods can be very efficient. These methods involve representing the system of equations as a matrix and then using techniques like Gaussian elimination or matrix inversion to solve for the variables. Matrix methods are often used in computer programs and are particularly well-suited for solving systems with many variables.

3. Graphical Method

For systems of two equations with two variables, the graphical method can be a visual way to find the solution. This method involves graphing each equation on the same coordinate plane. The point where the lines intersect represents the solution to the system. While this method is intuitive, it may not be accurate for systems with non-integer solutions or for systems with more than two variables.

Choosing the best method depends on the specific system of equations you're dealing with. The elimination method is generally a good choice for linear systems with three or more variables, while the substitution method can be effective when one equation is easily solved for a variable. Matrix methods are powerful for large systems, and the graphical method provides a visual understanding for systems with two variables. Experimenting with different methods will help you develop a strong intuition for solving systems of equations.

Conclusion: Mastering the Art of Solving Systems of Equations

Solving systems of equations is a fundamental skill in mathematics with applications across various fields. We've explored the elimination method in detail, providing a step-by-step guide to solving a system of three equations with three variables. We've also discussed the importance of verification and touched upon alternative methods like substitution, matrix methods, and graphical methods.

The key to mastering the art of solving systems of equations is practice. The more you work through different problems, the more comfortable you'll become with the techniques and the better you'll be able to identify the most efficient approach for a given system. Don't be afraid to make mistakes; they're valuable learning opportunities. Embrace the challenge, and you'll find that solving systems of equations can be a rewarding and empowering experience.

Remember, mathematics is not just about memorizing formulas; it's about developing problem-solving skills and logical thinking. By mastering techniques like solving systems of equations, you're not just learning math; you're honing your ability to analyze problems, develop strategies, and find solutions – skills that will serve you well in all aspects of life. So, keep practicing, keep exploring, and keep pushing your mathematical boundaries!