Solving Systems Of Equations A Step By Step Guide

Jul 9, 2025 by ADMIN 50 views

Solving systems of equations is a fundamental concept in mathematics with applications spanning various fields, including engineering, physics, economics, and computer science. This comprehensive guide delves into the step-by-step process of solving a system of three linear equations with three unknowns. Specifically, we will address the system:

-2x + y - z = -1
x + 3y + z = -6
2x - y - 2z = -2

This article will provide a detailed walkthrough of the solution, ensuring clarity and understanding for readers of all backgrounds. We'll explore the method of elimination, a powerful technique for simplifying systems of equations and isolating variables. By mastering this method, you'll gain a valuable tool for tackling similar problems in mathematics and beyond.

Understanding Systems of Equations

Before diving into the solution, it's crucial to grasp the underlying concept of systems of equations. A system of equations is a set of two or more equations containing the same variables. The solution to a system of equations is a set of values for the variables that satisfy all equations simultaneously. In the case of linear equations, each equation represents a straight line, and the solution corresponds to the point(s) where the lines intersect.

Our system involves three linear equations with three unknowns (x, y, and z). Each equation represents a plane in three-dimensional space, and the solution, if it exists, corresponds to the point where all three planes intersect. This intersection can be a single point, a line (indicating infinitely many solutions), or no intersection at all (indicating no solution).

Linear Equations and Their Significance

Linear equations form the bedrock of numerous mathematical models and real-world applications. Their simplicity and predictability make them invaluable tools for representing relationships between variables. In fields like economics, linear equations can model supply and demand curves, while in physics, they can describe the motion of objects under constant acceleration. Understanding how to solve systems of linear equations is therefore essential for anyone working with quantitative models.

The key characteristic of a linear equation is that the variables are raised to the power of 1. This ensures that the graph of the equation is a straight line (or a plane in higher dimensions). Non-linear equations, on the other hand, involve variables raised to higher powers or other non-linear functions (e.g., trigonometric or exponential functions), leading to curved graphs and more complex solution methods.

Methods for Solving Systems of Equations

Several methods exist for solving systems of equations, each with its own strengths and weaknesses. The most common methods include:

Substitution: This method involves solving one equation for one variable and substituting that expression into the other equation(s). This reduces the number of variables and equations, eventually leading to a solution.
Elimination: This method involves adding or subtracting multiples of equations to eliminate one or more variables. This simplifies the system and allows for easier solution of the remaining variables.
Matrix Methods: For larger systems of equations, matrix methods like Gaussian elimination or matrix inversion offer a more systematic and efficient approach.
Graphical Methods: For systems of two equations with two variables, graphical methods can be used to visualize the solution as the intersection point of the lines represented by the equations.

In this article, we will focus on the elimination method, as it is particularly well-suited for the system of equations we are addressing.

The Elimination Method: A Detailed Walkthrough

The elimination method is a powerful technique for solving systems of equations by strategically adding or subtracting multiples of equations to eliminate variables. This process simplifies the system, making it easier to solve for the remaining variables.

The core principle behind the elimination method lies in the fact that adding or subtracting equal quantities from both sides of an equation does not change its solution. By carefully choosing the multiples to add or subtract, we can eliminate specific variables, reducing the complexity of the system.

Let's apply the elimination method to our system of equations:

-2x + y - z = -1   (Equation 1)
x + 3y + z = -6    (Equation 2)
2x - y - 2z = -2   (Equation 3)

Step 1: Eliminate 'x' from Equations 2 and 3

Our first goal is to eliminate the variable 'x' from Equations 2 and 3. To achieve this, we can multiply Equation 2 by 2 and add it to Equation 1. This will cancel out the 'x' terms.

Multiply Equation 2 by 2: 2(x + 3y + z) = 2(-6) => 2x + 6y + 2z = -12

Add the modified Equation 2 to Equation 1:

(-2x + y - z) + (2x + 6y + 2z) = -1 + (-12)
7y + z = -13   (Equation 4)

Now, let's eliminate 'x' from Equation 3. We can directly add Equation 1 and Equation 3, as the 'x' terms have opposite signs:

Add Equation 1 and Equation 3:

(-2x + y - z) + (2x - y - 2z) = -1 + (-2)
-3z = -3

This simplifies to:

z = 1 (Equation 5) ```

Step 2: Solve for 'y' using Equations 4 and 5

Now that we have the value of 'z' (z = 1), we can substitute it into Equation 4 to solve for 'y':

Substitute z = 1 into Equation 4:

7y + 1 = -13
7y = -14
y = -2    (Equation 6)

Step 3: Solve for 'x' using Equations 2, 5, and 6

We now have the values of 'y' and 'z'. We can substitute these values into any of the original equations to solve for 'x'. Let's use Equation 2:

Substitute y = -2 and z = 1 into Equation 2:

x + 3(-2) + 1 = -6
x - 6 + 1 = -6
x - 5 = -6
x = -1

Step 4: Verify the Solution

It's crucial to verify our solution by substituting the values of x, y, and z into all three original equations to ensure they are satisfied:

Equation 1: -2(-1) + (-2) - 1 = 2 - 2 - 1 = -1 (Satisfied)
Equation 2: (-1) + 3(-2) + 1 = -1 - 6 + 1 = -6 (Satisfied)
Equation 3: 2(-1) - (-2) - 2(1) = -2 + 2 - 2 = -2 (Satisfied)

Since the solution satisfies all three equations, we can confidently state that the solution to the system is x = -1, y = -2, and z = 1.

The Solution: x = -1, y = -2, z = 1

Through the methodical application of the elimination method, we have successfully solved the system of equations:

-2x + y - z = -1
x + 3y + z = -6
2x - y - 2z = -2

The unique solution to this system is x = -1, y = -2, and z = 1. This solution represents the point of intersection of the three planes defined by the equations in three-dimensional space.

Understanding the Uniqueness of the Solution

In this case, we found a unique solution, indicating that the three planes intersect at a single point. However, it's important to recognize that systems of linear equations can have different types of solutions:

Unique Solution: As in our example, the system has one and only one solution.
Infinitely Many Solutions: The planes intersect along a line, meaning there are infinitely many points that satisfy all equations.
No Solution: The planes do not intersect at any point, indicating that there is no solution that satisfies all equations simultaneously.

The nature of the solution depends on the relationships between the equations in the system. If the equations are linearly independent (i.e., no equation can be expressed as a linear combination of the others), the system will typically have a unique solution. However, if the equations are linearly dependent, the system may have infinitely many solutions or no solution.

Alternative Methods and Their Applicability

While we employed the elimination method in this example, other methods, such as substitution or matrix methods, could also be used to solve this system. The choice of method often depends on the specific characteristics of the system and personal preference.

Substitution: The substitution method is particularly useful when one of the equations can be easily solved for one variable in terms of the others. However, for larger systems, the substitution method can become cumbersome.
Matrix Methods: Matrix methods, such as Gaussian elimination or matrix inversion, are highly efficient for solving large systems of equations. These methods involve representing the system in matrix form and applying systematic operations to the matrix to find the solution.

The Importance of Verification

As demonstrated in our solution process, verifying the solution is a crucial step. Substituting the obtained values back into the original equations ensures that the solution is correct and that no errors were made during the solution process. This step is particularly important when dealing with complex systems of equations where the likelihood of making a mistake is higher.

Conclusion: Mastering Systems of Equations

Solving systems of equations is a fundamental skill in mathematics with wide-ranging applications. This article has provided a detailed, step-by-step guide to solving a system of three linear equations with three unknowns using the elimination method. By understanding the principles behind the method and practicing its application, you can confidently tackle similar problems and gain a deeper appreciation for the power of linear algebra.

The ability to solve systems of equations is not only essential for mathematical problem-solving but also for modeling and analyzing real-world phenomena. From engineering design to economic forecasting, systems of equations provide a framework for understanding and predicting complex relationships between variables. By mastering this skill, you'll equip yourself with a valuable tool for success in various academic and professional pursuits. Remember to practice regularly and explore different methods to enhance your problem-solving abilities in this critical area of mathematics. The journey of mathematical understanding is continuous, and each solved problem brings you closer to mastery.