Solving Systems Of Equations A Comprehensive 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 satisfy all equations simultaneously. This article delves into the methods for solving systems of equations, providing a step-by-step guide with examples to enhance understanding. We will particularly focus on solving the given system of equations and verifying the correct solution. Let's explore the fascinating world of systems of equations and equip ourselves with the tools to tackle them effectively.

Understanding Systems of Equations

At its core, a system of equations represents a collection of mathematical relationships between variables. These relationships can be linear, quadratic, exponential, or any other form. The goal in solving a system of equations is to find the values of the variables that make all the equations in the system true at the same time. Think of it as finding the point(s) where the graphs of the equations intersect. This intersection point represents the solution that satisfies all equations. Systems of equations can have one solution, no solution, or infinitely many solutions, depending on the nature of the equations and their relationships. A system with one solution means the lines intersect at a single point. A system with no solution means the lines are parallel and never intersect. A system with infinitely many solutions means the lines are the same, overlapping at every point. Understanding these possibilities is crucial for interpreting the results of solving systems of equations and applying them to real-world scenarios. The techniques we use to solve these systems are varied and powerful, offering a range of approaches to suit different types of equations and complexities. So, as we dive deeper into the methods, remember that each method offers a unique perspective on how to untangle these mathematical relationships and reveal the underlying solutions.

Methods for Solving Systems of Equations

There are several methods for solving systems of equations, each with its advantages and disadvantages. The most common methods include:

1. Substitution: This method involves solving one equation for one variable and substituting that expression into the other equation. This eliminates one variable, allowing you to solve for the remaining variable. Once you find the value of one variable, you can substitute it back into either of the original equations to find the value of the other variable. This method is particularly useful when one of the equations can be easily solved for one variable in terms of the other. For example, if you have an equation like y = 2x + 3, substitution is a straightforward approach. However, for more complex equations, this method can sometimes become cumbersome, involving complicated algebraic manipulations. The key to success with substitution is to carefully track each step and ensure accurate substitutions to avoid errors. By mastering this technique, you gain a powerful tool for simplifying and solving a wide range of systems of equations.
2. Elimination (or Addition/Subtraction): This method involves manipulating the equations so that the coefficients of one variable are opposites. Then, you add the equations together, which eliminates that variable. This leaves you with a single equation in one variable, which you can solve. Once you find the value of that variable, you can substitute it back into either of the original equations to find the value of the other variable. The elimination method shines when the equations are structured in a way that makes it easy to cancel out variables. For instance, if you have equations with terms like +2y and -2y, adding the equations directly will eliminate y. This method is particularly efficient when the coefficients of one variable are already opposites or can be easily made opposites by multiplying one or both equations by a constant. It's a systematic approach that reduces the complexity of the system step by step. With practice, you'll be able to quickly identify situations where elimination is the most efficient method, making it an invaluable tool in your problem-solving arsenal.
3. Graphing: This method involves graphing each equation on the same coordinate plane. The solution to the system is the point(s) where the graphs intersect. Graphing is a visual method that can provide a clear understanding of the solutions to a system of equations. When you graph each equation, the intersection points represent the values of the variables that satisfy both equations simultaneously. This method is especially useful for linear equations, as their graphs are straight lines, making the intersection points easy to identify. However, graphing can be less precise for complex equations or when the solutions are not integers. Additionally, it may not be practical for systems with more than two variables. Despite these limitations, graphing offers a valuable visual aid and can be a great way to check solutions obtained through algebraic methods. It helps to build intuition about the relationship between equations and their solutions, making it a fundamental technique in understanding systems of equations.
4. Matrix Methods: This method involves representing the system of equations as a matrix and using matrix operations to solve for the variables. Matrix methods are particularly useful for solving systems with three or more variables. When dealing with large systems of equations, matrix methods provide a systematic and efficient approach. The process involves transforming the system into matrix form, then applying techniques such as Gaussian elimination or matrix inversion to find the solution. These methods are especially powerful because they can be easily implemented using computer software, making them practical for real-world problems with many variables. While the underlying concepts may seem abstract at first, mastering matrix methods opens up a whole new level of problem-solving capability. They are widely used in various fields, including engineering, physics, and economics, to model and solve complex systems. As you delve deeper into mathematics and its applications, you'll find that matrix methods are an indispensable tool for tackling advanced problems.

Solving the Given System of Equations

The given system of equations is:

$\begin{cases} 3x + 2y = 39 \\ 5x - y = 13 \end{cases}$

We will use the elimination method to solve this system. First, we multiply the second equation by 2 to make the coefficients of y opposites:

$2(5x - y) = 2(13) \Rightarrow 10x - 2y = 26$

Now our system of equations is:

$\begin{cases} 3x + 2y = 39 \\ 10x - 2y = 26 \end{cases}$

Next, we add the two equations together to eliminate y:

$(3x + 2y) + (10x - 2y) = 39 + 26 \Rightarrow 13x = 65$

Now, we solve for x:

$x = \frac{65}{13} = 5$

Now that we have the value of x, we can substitute it back into either of the original equations to find the value of y. We will use the first equation:

$3(5) + 2y = 39 \Rightarrow 15 + 2y = 39$

Subtract 15 from both sides:

$2y = 39 - 15 \Rightarrow 2y = 24$

Divide by 2 to solve for y:

$y = \frac{24}{2} = 12$

Therefore, the solution to the system of equations is (x, y) = (5, 12).

Verifying the Solution

To ensure our solution is correct, we must substitute the values of x and y back into both original equations and verify that they hold true. This step is crucial in problem-solving, as it helps catch any potential errors made during the solution process. By plugging in the values, we can confirm whether the solution satisfies all the equations in the system, giving us confidence in our answer. This verification process not only ensures accuracy but also reinforces our understanding of how the variables interact within the equations. It's a simple yet powerful way to double-check our work and solidify our grasp of the concepts involved. So, let's move forward and meticulously verify our solution to ensure its validity and our comprehension of the problem.

Verification Process

Let's verify the solution (5, 12) for the given system of equations:

1. First Equation: 3x + 2y = 39

   Substitute x = 5 and y = 12:

   3(5) + 2(12) = 15 + 24 = 39

   The first equation holds true.

2. Second Equation: 5x - y = 13

   Substitute x = 5 and y = 12:

   5(5) - 12 = 25 - 12 = 13

   The second equation also holds true.

Since the solution (5, 12) satisfies both equations, we can confidently say that it is the correct solution to the system of equations.

Conclusion

In this comprehensive guide, we've explored the fundamentals of solving systems of equations, focusing on the elimination method to solve a specific problem. The solution to the system of equations is (5, 12), which we verified by substituting the values back into the original equations. Understanding and mastering the techniques for solving systems of equations is crucial for success in mathematics and its applications. By grasping these concepts, you equip yourself with a powerful tool for tackling a wide range of problems in various fields. Remember, practice is key to proficiency, so keep exploring different systems of equations and honing your skills. The journey of mathematical discovery is a rewarding one, and with each solved problem, you strengthen your ability to think critically and solve complex challenges. So, embrace the challenge, and continue your exploration of the fascinating world of mathematics!