Solving Systems Of Equations A Step-by-Step Guide

In the realm of mathematics, solving systems of equations is a fundamental skill with applications spanning various fields, from engineering to economics. This article delves into the process of solving a specific system of linear equations, providing a step-by-step guide and explanations to enhance your understanding. Let's embark on this mathematical journey together.

The Given System of Equations

We are presented with the following system of equations:

$\begin{aligned}
x + 2y &= 10 \\
-x - 5y &= -19
$\end{aligned}

Our goal is to find the values of x and y that satisfy both equations simultaneously. There are several methods to accomplish this, including substitution, elimination, and matrix methods. In this article, we will employ the elimination method, which is particularly efficient for this system.

The Elimination Method: A Detailed Approach

The elimination method, also known as the addition method, involves manipulating the equations in the system to eliminate one variable, thereby allowing us to solve for the other. This is achieved by adding or subtracting the equations in such a way that the coefficients of one variable cancel each other out. In our given system, the coefficients of x are already opposites (1 and -1), making the elimination method a natural choice.

Step 1: Adding the Equations

We begin by adding the two equations together:

(x + 2y) + (-x - 5y) = 10 + (-19)

Simplifying the equation, we get:

x - x + 2y - 5y = -9

-3y = -9

Notice that the x terms have canceled out, leaving us with a single equation in terms of y.

Step 2: Solving for y

To isolate y, we divide both sides of the equation by -3:

-3y / -3 = -9 / -3

y = 3

Thus, we have found the value of y to be 3.

Step 3: Substituting y to Find x

Now that we know y = 3, we can substitute this value into either of the original equations to solve for x. Let's use the first equation:

x + 2y = 10

Substituting y = 3, we get:

x + 2(3) = 10

x + 6 = 10

Step 4: Solving for x

To isolate x, we subtract 6 from both sides of the equation:

x + 6 - 6 = 10 - 6

x = 4

Therefore, the value of x is 4.

The Solution

We have successfully solved the system of equations using the elimination method. The solution is:

x = 4
y = 3

This means that the point (4, 3) is the intersection of the two lines represented by the given equations. To verify our solution, we can substitute these values back into the original equations.

Verification

Let's substitute x = 4 and y = 3 into the first equation:

x + 2y = 10

4 + 2(3) = 10

4 + 6 = 10

10 = 10

The first equation holds true. Now, let's substitute the values into the second equation:

-x - 5y = -19

-4 - 5(3) = -19

-4 - 15 = -19

-19 = -19

The second equation also holds true. Since the solution satisfies both equations, we can confidently conclude that our solution is correct.

Alternative Methods for Solving Systems of Equations

While we used the elimination method in this example, it's important to be aware of other methods for solving systems of equations. These include:

1. Substitution Method: This method involves solving one equation for one variable and substituting that expression into the other equation. This results in a single equation with one variable, which can then be solved. The solution is then substituted back into one of the original equations to find the value of the other variable.

2. Graphical Method: This method involves graphing both equations on a coordinate plane. The point of intersection of the two lines represents the solution to the system. This method is particularly useful for visualizing the solution and understanding the relationship between the equations.

3. Matrix Methods: For larger systems of equations, matrix methods such as Gaussian elimination or using the inverse of a matrix can be more efficient. These methods involve representing the system of equations in matrix form and then performing operations on the matrices to solve for the variables.

The choice of method depends on the specific system of equations and the preference of the solver. For simple systems like the one we solved, elimination or substitution are often the most straightforward methods.

Applications of Systems of Equations

Systems of equations are not just abstract mathematical concepts; they have numerous real-world applications. Here are a few examples:

• Engineering: Systems of equations are used to analyze circuits, solve structural problems, and model fluid flow.
• Economics: They are used to model supply and demand, analyze market equilibrium, and make economic forecasts.
• Computer Graphics: Systems of equations are used in computer graphics to perform transformations, such as rotations and scaling, and to create realistic images.
• Cryptography: They are used in cryptography to encrypt and decrypt messages.
• Everyday Life: They can be used to solve problems involving mixtures, rates, and distances.

Conclusion

Solving systems of equations is a fundamental skill in mathematics with broad applications. In this article, we demonstrated the elimination method to solve a specific system of linear equations, providing a detailed step-by-step explanation. We also discussed alternative methods and highlighted the real-world applications of systems of equations. By mastering these techniques, you will be well-equipped to tackle a wide range of mathematical and practical problems.

Remember, practice is key to developing proficiency in solving systems of equations. Work through various examples, explore different methods, and don't hesitate to seek help when needed. With dedication and perseverance, you can unlock the power of systems of equations and apply them to solve real-world challenges.

This comprehensive guide aimed to provide a clear and thorough understanding of solving systems of equations. By understanding the concepts and methods discussed, you'll be well-prepared to tackle more complex mathematical problems and appreciate the versatility of these techniques in various fields.