Solving Systems Of Linear Equations A Step By Step Guide

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In the realm of mathematics, particularly in algebra, systems of linear equations play a crucial role. These systems represent a set of two or more linear equations that share the same variables. The solution to a system of linear equations is the set of values for the variables that satisfy all equations simultaneously. In simpler terms, it's the point where the lines represented by the equations intersect on a graph. Understanding how to solve these systems is fundamental for various applications, from simple problem-solving to complex modeling in fields like economics, engineering, and computer science. This article will guide you through the process of solving a specific system of linear equations and provide a comprehensive understanding of the methods involved.

Understanding the Problem

Before diving into the solution, let's clearly state the problem we're addressing. We are given the following system of linear equations:

2x+y=13x−y=−6 \begin{array}{l} 2 x+y=1 \\ 3 x-y=-6 \end{array}

Our goal is to find the values of x and y that satisfy both equations. To do this, we'll explore several methods commonly used to solve systems of linear equations. These methods include graphing, substitution, and elimination. Each method has its advantages and disadvantages, and the best approach often depends on the specific equations in the system. For this particular system, we'll focus on the elimination method, which is particularly efficient when the coefficients of one of the variables are opposites or can be easily made opposites.

Methods for Solving Systems of Linear Equations

1. Graphing

Graphing is a visual method for solving systems of linear equations. Each equation in the system represents a line on a coordinate plane. The solution to the system is the point where the lines intersect. To solve by graphing, you first graph each equation on the same coordinate plane. The point of intersection, if one exists, represents the solution to the system. If the lines are parallel, there is no solution. If the lines coincide, there are infinitely many solutions. While graphing provides a clear visual representation of the solution, it can be less precise than algebraic methods, especially when the solution involves non-integer values.

2. Substitution

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This process results in a single equation with one variable, which can then be solved. Once you've found the value of one variable, you can substitute it back into either of the original equations to find the value of the other variable. The substitution method is particularly useful when one of the equations is already solved for one variable or when it's easy to isolate one variable.

3. Elimination

The elimination method, also known as the addition method, involves manipulating the equations so that the coefficients of one of the variables are opposites. When the equations are added together, that variable is eliminated, leaving a single equation with one variable. This equation can then be solved, and the value can be substituted back into one of the original equations to find the value of the other variable. The elimination method is highly effective when the coefficients of one of the variables are already opposites or can be easily made opposites by multiplying one or both equations by a constant.

Solving the System Using Elimination

For the given system of equations:

2x+y=13x−y=−6 \begin{array}{l} 2 x+y=1 \\ 3 x-y=-6 \end{array}

We can see that the coefficients of y are already opposites (1 and -1). This makes the elimination method the most straightforward approach. Here's how we apply it:

  1. Add the two equations together:

(2x+y)+(3x−y)=1+(−6)(2x + y) + (3x - y) = 1 + (-6)

  1. Simplify the equation:

    5x=−55x = -5

  2. Solve for x:

    x=−1x = -1

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

  1. Substitute x = -1 into the first equation:

    2(−1)+y=12(-1) + y = 1

  2. Simplify and solve for y:

    −2+y=1-2 + y = 1

    y=3y = 3

Therefore, the solution to the system of equations is x = -1 and y = 3. This can be written as the ordered pair (-1, 3).

Verifying the Solution

It's always a good practice to verify the solution by substituting the values of x and y back into both original equations to ensure they are satisfied. Let's do that:

  1. First equation:

    2x+y=12x + y = 1

    2(−1)+3=12(-1) + 3 = 1

    −2+3=1-2 + 3 = 1

    1=11 = 1 (The equation is satisfied)

  2. Second equation:

    3x−y=−63x - y = -6

    3(−1)−3=−63(-1) - 3 = -6

    −3−3=−6-3 - 3 = -6

    −6=−6-6 = -6 (The equation is satisfied)

Since the values x = -1 and y = 3 satisfy both equations, we can confidently conclude that (-1, 3) is the correct solution to the system of linear equations.

Analyzing the Answer Choices

We were given the following answer choices:

  • A. (-1, 3)
  • B. (1, -1)
  • C. (2, 3)
  • D. (5, 0)

Based on our calculations and verification, the correct answer is A. (-1, 3). The other options do not satisfy both equations in the system.

Importance of Solving Systems of Linear Equations

The ability to solve systems of linear equations is a fundamental skill in mathematics with far-reaching applications. These systems arise in various real-world scenarios, making their solutions essential for practical problem-solving. Here are a few key reasons why understanding and mastering the solution of systems of linear equations is important:

  1. Modeling Real-World Problems: Systems of linear equations are often used to model real-world situations involving multiple variables and constraints. For example, in economics, they can be used to model supply and demand relationships. In physics, they can describe the motion of objects under certain forces. In engineering, they are used in circuit analysis and structural design.

  2. Optimization: Many optimization problems, where the goal is to maximize or minimize a certain quantity subject to constraints, can be formulated as systems of linear equations or inequalities. Linear programming, a powerful optimization technique, relies heavily on solving such systems.

  3. Data Analysis: In data analysis and statistics, systems of linear equations are used in regression analysis, where the goal is to find the best-fitting linear relationship between variables. They are also used in solving problems related to matrices and vectors, which are fundamental tools in data science and machine learning.

  4. Computer Graphics: Systems of linear equations play a crucial role in computer graphics for tasks such as transformations (e.g., rotations, scaling, and translations) and projections. They are used to determine how objects should be displayed on a screen.

  5. Foundation for Advanced Mathematics: The concepts and techniques used to solve systems of linear equations serve as a foundation for more advanced mathematical topics, such as linear algebra, differential equations, and numerical analysis. A strong understanding of these systems is essential for further studies in mathematics and related fields.

Conclusion

In conclusion, solving systems of linear equations is a fundamental skill in mathematics with wide-ranging applications. We successfully solved the given system:

2x+y=13x−y=−6 \begin{array}{l} 2 x+y=1 \\ 3 x-y=-6 \end{array}

Using the elimination method, we found that the solution is x = -1 and y = 3, which corresponds to the ordered pair (-1, 3). This solution satisfies both equations in the system. We also discussed the importance of verifying the solution and explored various methods for solving systems of linear equations, including graphing, substitution, and elimination.

By understanding the methods and applications of solving systems of linear equations, you'll be well-equipped to tackle a wide range of mathematical and real-world problems. The ability to solve these systems is a valuable asset in various fields, from science and engineering to economics and computer science. Continue practicing and exploring different types of systems to further enhance your skills and deepen your understanding of this important mathematical concept.

To reinforce your understanding, consider exploring additional examples and exercises. Practice applying different methods, such as substitution and graphing, to solve various systems of linear equations. Understanding the strengths and weaknesses of each method will help you choose the most efficient approach for a given problem. Furthermore, consider exploring real-world applications of systems of linear equations to appreciate their practical significance. By actively engaging with the material and seeking out opportunities for application, you can solidify your understanding and develop a strong foundation in this essential mathematical topic. Remember, mastery comes with practice, so keep exploring and challenging yourself! The more you work with systems of linear equations, the more comfortable and confident you will become in solving them.

This foundational knowledge will not only benefit you in your academic pursuits but also equip you with valuable problem-solving skills applicable to a wide range of real-world scenarios. Systems of linear equations are not just abstract mathematical concepts; they are powerful tools that can help you understand and model the world around you. Embrace the challenge, delve deeper into the topic, and unlock the potential of this essential mathematical tool.