Solving Systems Of Equations A Step By Step Guide

JU07/23/2025 08, 2025 by THE IDEN 50 views

Solving a System of Equations: A Step-by-Step Guide

In mathematics, solving systems of equations is a fundamental skill with applications across various fields, from engineering and physics to economics and computer science. A system of equations is a set of two or more equations containing the same variables. The solution to a system of equations is the set of values for the variables that make all the equations in the system true simultaneously. This article provides a comprehensive guide to solving a specific system of equations, focusing on the techniques and steps involved in finding the solution. Understanding these methods is crucial for anyone dealing with mathematical modeling, data analysis, or any area that requires the manipulation of multiple variables and equations. The ability to solve systems of equations efficiently and accurately is not just an academic exercise; it is a practical skill that enhances problem-solving capabilities in numerous real-world scenarios. By mastering these techniques, individuals can gain a deeper understanding of mathematical relationships and their applications, fostering a more analytical and solution-oriented approach to complex problems.

Problem Statement

We are given the following system of equations:

y + 3x = e^(-2t)
x - 3y = e^(2t)

with initial conditions:

t = 0, y = 0, x = 0

Our goal is to solve this system for x and y as functions of t. This involves finding expressions for x(t) and y(t) that satisfy both equations simultaneously, while also adhering to the given initial conditions. The process of solving this system will require a combination of algebraic manipulation and techniques for handling exponential functions. The initial conditions provide a starting point for finding a particular solution, making the problem more specific and applicable to scenarios where initial states are known. Understanding the interplay between the equations and the initial conditions is key to successfully navigating the solution process. This problem is a classic example of how systems of equations arise in various contexts, particularly in dynamic systems where variables change over time. The solution will not only give us the mathematical expressions for x and y but also a deeper insight into how these variables relate to each other and evolve with changes in t. By the end of this guide, you will have a clear understanding of the steps required to solve this system and a solid foundation for tackling similar problems.

Step 1: Elimination Method

The elimination method is a technique used to solve systems of equations by eliminating one variable, making it easier to solve for the other. In this case, we can eliminate y by manipulating the two equations:

Multiply the first equation by 3:

3(y + 3x) = 3e^(-2t)
3y + 9x = 3e^(-2t)

Add the modified first equation to the second equation:

(3y + 9x) + (x - 3y) = 3e^(-2t) + e^(2t)
10x = 3e^(-2t) + e^(2t)

Solve for x:
```
x = (3e^(-2t) + e^(2t)) / 10
```

The elimination method is a powerful tool for simplifying complex systems of equations. By strategically multiplying equations and adding them together, we can cancel out variables and reduce the system to a single equation with a single unknown. This makes the solution process much more manageable. In this specific instance, eliminating y allowed us to isolate x and express it as a function of t. The resulting equation for x is a combination of exponential terms, reflecting the nature of the original system. This step is crucial because it breaks down the initial problem into a more solvable form, paving the way for determining the value of the remaining variable, y. The elimination method's effectiveness lies in its ability to transform the system into a more transparent structure, highlighting the relationships between variables and making the solution pathway clearer. Mastering this technique is essential for anyone working with systems of equations, as it provides a systematic approach to finding solutions in a variety of mathematical and scientific contexts. The application of the elimination method here demonstrates its versatility and underscores its importance in solving complex problems. This methodical approach not only helps in finding the solution but also enhances the understanding of the underlying mathematical principles.

Step 2: Solve for y

Now that we have x as a function of t, we can substitute it back into either of the original equations to solve for y. Let's use the first equation:

y + 3x = e^(-2t)

Substitute the expression for x:

y + 3((3e^(-2t) + e^(2t)) / 10) = e^(-2t)

Solve for y:

y = e^(-2t) - (9e^(-2t) + 3e^(2t)) / 10

y = (10e^(-2t) - 9e^(-2t) - 3e^(2t)) / 10

y = (e^(-2t) - 3e^(2t)) / 10

Solving for y involves a process of substitution and simplification that builds upon the previous step of finding x. By plugging the expression for x into one of the original equations, we create an equation that only contains y and t. This step is critical because it allows us to isolate y and express it as a function of t. The algebraic manipulations required in this step, such as distributing coefficients and combining like terms, are fundamental skills in solving mathematical problems. The resulting expression for y is a combination of exponential functions, similar to x, which indicates the interconnected nature of the variables in the system. This step not only provides the solution for y but also reinforces the understanding of how variables interact within a system of equations. The ability to manipulate equations and solve for unknowns is a cornerstone of mathematical problem-solving, and this step exemplifies the application of these skills in a practical context. The process of finding y highlights the importance of careful algebraic manipulation and attention to detail, ensuring that the final expression accurately represents the relationship between y and t. This thorough approach is essential for achieving correct solutions and building confidence in mathematical abilities.

Step 3: Apply Initial Conditions

The initial conditions given are t = 0, x = 0, and y = 0. Let's check if our solutions satisfy these conditions:

For x:

x(0) = (3e^(0) + e^(0)) / 10 = (3 + 1) / 10 = 4 / 10 = 2 / 5

For y:

y(0) = (e^(0) - 3e^(0)) / 10 = (1 - 3) / 10 = -2 / 10 = -1 / 5

We observe that the initial conditions are not satisfied by our current solutions. This indicates that there might be an error in our calculations, or the initial conditions may not be consistent with the system of equations. Let's re-examine our steps to identify any potential mistakes.

Applying initial conditions is a crucial step in verifying the correctness of a solution to a system of equations, especially when dealing with differential equations or dynamic systems. The initial conditions provide specific values for the variables at a particular point in time, allowing us to check if our derived solutions align with these known values. In this case, the initial conditions t = 0, x = 0, and y = 0 serve as a benchmark against which we can test our solutions for x(t) and y(t). When we substitute t = 0 into our expressions for x and y, we obtain x(0) = 2/5 and y(0) = -1/5, which do not match the given initial conditions x = 0 and y = 0. This discrepancy signals a potential issue in our solution process, prompting a thorough review of our calculations and steps. The act of checking initial conditions is not merely a formality; it is an integral part of the problem-solving process that helps us identify errors and ensure the accuracy of our results. This step underscores the importance of precision and attention to detail in mathematical problem-solving. By recognizing the mismatch between our solutions and the initial conditions, we are able to refine our approach and strive for a more accurate solution. This iterative process of solving, checking, and refining is fundamental to mathematical inquiry and problem-solving.

Step 4: Re-evaluation and Correct Solution

Upon reviewing the calculations, we realize that the initial conditions x = 0 and y = 0 at t = 0 are not directly applicable to the general solution of the given system of equations. The system is a set of algebraic equations, not a differential equation, and the initial conditions seem inconsistent with the equations themselves.

Let's focus on the general solution we derived:

x = (3e^(-2t) + e^(2t)) / 10
y = (e^(-2t) - 3e^(2t)) / 10

These are the correct general solutions for x and y as functions of t for the given system of equations. The initial conditions provided do not seem to play a role in finding this general solution, as they would typically be used to solve for constants in the solution of a differential equation, which is not the case here.

Re-evaluation and correction are essential components of the problem-solving process, particularly in mathematics. After applying initial conditions and finding a discrepancy, it becomes necessary to step back and critically examine each step of the solution. This process involves not only checking the calculations for errors but also reassessing the applicability of the methods used and the interpretation of the problem statement. In this instance, the re-evaluation revealed that the initial conditions x = 0 and y = 0 at t = 0 are inconsistent with the general solution of the given algebraic system of equations. This realization is crucial because it highlights the importance of understanding the nature of the problem and the appropriate techniques to apply. The system of equations is algebraic, not differential, and therefore, the initial conditions do not serve the typical purpose of determining specific constants in a general solution. Instead, they indicate a potential inconsistency within the problem statement itself. The act of correcting the solution involves focusing on the general solutions for x and y that were derived earlier, which are the valid expressions for these variables as functions of t. This step underscores the importance of flexibility and adaptability in problem-solving, as well as the ability to distinguish between different types of mathematical problems and their corresponding solution methods. The process of re-evaluation and correction not only leads to a more accurate solution but also deepens the understanding of the underlying mathematical concepts and principles.

Final Solution

The solution to the system of equations is:

x(t) = (3e^(-2t) + e^(2t)) / 10
y(t) = (e^(-2t) - 3e^(2t)) / 10

These equations represent the general solution for the given system. The initial conditions t = 0, x = 0, and y = 0 are not consistent with this system and do not affect the general solution.

The final solution encapsulates the culmination of the problem-solving process, representing the most accurate and complete answer to the given problem. In this case, the final solution consists of the expressions for x(t) and y(t), which describe the general relationship between the variables x, y, and t in the system of equations. These expressions are derived through the elimination method and algebraic manipulation, providing a clear and concise representation of the solution. The acknowledgment that the initial conditions are inconsistent with the system is a crucial aspect of the final solution, as it highlights a potential issue within the problem statement itself. The recognition that these initial conditions do not affect the general solution is also significant, as it demonstrates a deep understanding of the mathematical principles at play. The final solution not only provides the mathematical expressions for x and y but also offers a comprehensive understanding of the problem's context and the validity of the results. This level of understanding is essential for applying the solution in real-world scenarios and for building confidence in mathematical problem-solving abilities. The final solution serves as a testament to the methodical approach taken, the careful calculations performed, and the critical evaluation of the results. It represents the ultimate goal of the problem-solving process and underscores the importance of accuracy, precision, and thoroughness in mathematical work.