Solving Systems Of Equations X + Y = 6 And X + 3y = -2 A Step-by-Step Guide
Introduction to Systems of Equations
Systems of equations are a fundamental concept in algebra and are essential for solving a variety of real-world problems. This guide focuses on one such system: { x + y = 6 } and { x + 3y = -2 }. Solving systems of equations involves finding the values of the variables that satisfy all the equations simultaneously. This comprehensive guide will walk you through different methods to solve this specific system, providing clear, step-by-step explanations to ensure a thorough understanding. Mastering the techniques for solving systems of equations not only enhances your algebraic skills but also provides a powerful tool for tackling more complex mathematical problems and real-world applications. Whether you are a student learning these concepts for the first time or someone looking to refresh your knowledge, this article will provide a solid foundation. We will explore the substitution method, the elimination method, and graphical methods, highlighting the strengths and weaknesses of each approach. Understanding these methods will empower you to choose the most efficient technique for any given system of equations, enhancing your problem-solving skills and mathematical fluency. Solving systems of equations is not just an academic exercise; it is a practical skill used in various fields, including engineering, economics, computer science, and many others. By mastering this topic, you are equipping yourself with a valuable tool for both academic and professional pursuits. In the following sections, we will delve deeper into the specific system { x + y = 6 } and { x + 3y = -2 }, providing detailed solutions and explanations.
Understanding the Equations
Before diving into the solutions, it’s crucial to understand the equations we’re dealing with: x + y = 6 and x + 3y = -2. These are two linear equations, each representing a straight line on a graph. The solution to the system of equations is the point where these two lines intersect. Grasping this graphical representation can make the algebraic solutions more intuitive. Each equation contains two variables, x and y, and the goal is to find the unique pair of values (x, y) that satisfies both equations. The first equation, x + y = 6, represents a line with a slope of -1 and a y-intercept of 6. This means that for every one unit increase in x, y decreases by one unit, and the line crosses the y-axis at the point (0, 6). The second equation, x + 3y = -2, represents a line with a different slope and y-intercept. This line has a slope of -1/3 and a y-intercept of -2/3. Understanding these slopes and intercepts can help visualize how the two lines will intersect. The intersection point, which is the solution to the system, is the only point that lies on both lines simultaneously. This visual understanding is particularly useful when using graphical methods to solve systems of equations. By plotting the two lines on a coordinate plane, the intersection point can be visually identified, providing a geometric interpretation of the algebraic solution. Moreover, understanding the nature of these equations helps in predicting the type of solution we might encounter. In general, a system of two linear equations can have one solution (the lines intersect at a single point), no solutions (the lines are parallel and never intersect), or infinitely many solutions (the lines are identical and overlap completely). In the case of our system, { x + y = 6 } and { x + 3y = -2 }, we expect to find a unique solution because the lines have different slopes and are not parallel.
Method 1: Solving by Substitution
The substitution method is a powerful technique for solving systems of equations. It involves solving one equation for one variable and then substituting that expression into the other equation. For the system { x + y = 6 } and { x + 3y = -2 }, let's start by solving the first equation for x. From x + y = 6, we can isolate x by subtracting y from both sides, resulting in x = 6 - y. Now, we substitute this expression for x into the second equation, x + 3y = -2. Replacing x with (6 - y) gives us (6 - y) + 3y = -2. This new equation contains only one variable, y, which we can now solve. Simplifying the equation, we combine like terms: 6 + 2y = -2. Next, we subtract 6 from both sides to isolate the term with y: 2y = -8. Finally, we divide both sides by 2 to find the value of y: y = -4. Now that we have the value of y, we can substitute it back into the equation x = 6 - y to find the value of x. Substituting y = -4 gives us x = 6 - (-4), which simplifies to x = 6 + 4, so x = 10. Therefore, the solution to the system of equations is x = 10 and y = -4. This means that the point (10, -4) is the intersection point of the two lines represented by the equations. To verify our solution, we can substitute these values back into both original equations. For the first equation, x + y = 6, we have 10 + (-4) = 6, which is true. For the second equation, x + 3y = -2, we have 10 + 3(-4) = 10 - 12 = -2, which is also true. This confirms that our solution is correct. The substitution method is particularly useful when one of the equations can be easily solved for one variable in terms of the other. It transforms the system of two equations into a single equation with one variable, making it straightforward to solve. This method is applicable to a wide range of systems of equations, making it a valuable tool in algebra.
Method 2: Solving by Elimination
The elimination method, also known as the addition method, is another effective technique for solving systems of equations. This method involves manipulating the equations so that when they are added together, one of the variables is eliminated. For the system { x + y = 6 } and { x + 3y = -2 }, our goal is to eliminate either x or y. To eliminate x, we can multiply the first equation by -1. This gives us -x - y = -6. The second equation remains unchanged: x + 3y = -2. Now, we add the modified first equation to the second equation: (-x - y) + (x + 3y) = -6 + (-2). This simplifies to 2y = -8. Dividing both sides by 2, we find that y = -4. Next, we substitute this value of y back into one of the original equations to solve for x. Let's use the first equation, x + y = 6. Substituting y = -4 gives us x + (-4) = 6, which simplifies to x - 4 = 6. Adding 4 to both sides, we find that x = 10. Thus, the solution to the system of equations is x = 10 and y = -4, the same result we obtained using the substitution method. To verify our solution, we can substitute these values back into both original equations. For the first equation, x + y = 6, we have 10 + (-4) = 6, which is true. For the second equation, x + 3y = -2, we have 10 + 3(-4) = 10 - 12 = -2, which is also true. This confirms the correctness of our solution. The elimination method is especially useful when the coefficients of one of the variables are either the same or easily made the same (or opposite) by multiplication. By strategically adding or subtracting the equations, we can eliminate one variable and solve for the other. This method is a powerful alternative to substitution and is widely used in algebra. In summary, the elimination method provides a systematic approach to solving systems of equations by eliminating one variable, thereby simplifying the problem to a single equation with one unknown. This method, like substitution, is a fundamental technique in algebra and is essential for solving a wide range of mathematical problems.
Method 3: Solving Graphically
Solving systems of equations graphically provides a visual understanding of the solution. Each equation in the system represents a line on a coordinate plane, and the solution to the system is the point where the lines intersect. For the system { x + y = 6 } and { x + 3y = -2 }, we first need to graph each equation. To graph x + y = 6, we can find two points on the line. Let’s find the x-intercept and y-intercept. When y = 0, x = 6, so the x-intercept is (6, 0). When x = 0, y = 6, so the y-intercept is (0, 6). Plotting these points and drawing a line through them gives us the graph of the first equation. For the second equation, x + 3y = -2, we can again find two points. When y = 0, x = -2, so the x-intercept is (-2, 0). When x = 0, 3y = -2, so y = -2/3, and the y-intercept is (0, -2/3). Plotting these points and drawing a line through them gives us the graph of the second equation. The point where these two lines intersect is the solution to the system of equations. By carefully plotting the lines on a graph, we can visually estimate the intersection point. In this case, the lines intersect at the point (10, -4). This graphical solution confirms the algebraic solutions we found using the substitution and elimination methods. While graphical methods can be very intuitive, they are often less precise than algebraic methods, especially when the solution involves non-integer coordinates. However, graphing provides a valuable way to visualize the system of equations and understand the relationship between the equations and their solutions. It is particularly useful for understanding the concepts of consistent and inconsistent systems, as well as dependent and independent systems. A consistent system has at least one solution, which corresponds to the lines intersecting at one or more points. An inconsistent system has no solution, which means the lines are parallel and do not intersect. A dependent system has infinitely many solutions, which occurs when the two equations represent the same line. In our case, the system { x + y = 6 } and { x + 3y = -2 } is consistent and independent, as the lines intersect at a single point, indicating a unique solution.
Verifying the Solution
After solving a system of equations, it is crucial to verify the solution to ensure accuracy. This step helps catch any errors made during the solving process and confirms that the values obtained for the variables satisfy all the equations in the system. We found the solution to the system { x + y = 6 } and { x + 3y = -2 } to be x = 10 and y = -4. To verify this solution, we substitute these values back into the original equations. For the first equation, x + y = 6, we substitute x = 10 and y = -4, giving us 10 + (-4) = 6. This simplifies to 6 = 6, which is a true statement. This confirms that the solution satisfies the first equation. For the second equation, x + 3y = -2, we substitute x = 10 and y = -4, giving us 10 + 3(-4) = -2. This simplifies to 10 - 12 = -2, which further simplifies to -2 = -2. This is also a true statement, confirming that the solution satisfies the second equation. Since the values x = 10 and y = -4 satisfy both equations in the system, we can confidently conclude that this is the correct solution. Verifying the solution is a best practice that should be followed whenever solving systems of equations. It not only ensures accuracy but also reinforces understanding of the concept. By substituting the values back into the original equations, we are essentially checking that the solution lies on both lines represented by the equations. This provides a concrete connection between the algebraic solution and the graphical representation of the system. In addition to catching errors, verification also helps in building confidence in the solution. It provides a sense of closure and ensures that the problem has been solved correctly. This is particularly important in exams and assessments, where accuracy is paramount. In summary, verifying the solution is an essential step in the process of solving systems of equations. It ensures accuracy, reinforces understanding, and builds confidence in the final answer.
Real-World Applications
Systems of equations are not just theoretical mathematical concepts; they have numerous real-world applications across various fields. Understanding how to apply these concepts can significantly enhance problem-solving skills in practical situations. One common application is in economics, where systems of equations are used to model supply and demand. For instance, the price and quantity of a product can be determined by solving a system of equations representing the supply and demand curves. Similarly, in business, systems of equations can be used to optimize resource allocation, calculate break-even points, and analyze profit margins. Consider a scenario where a company produces two products, A and B. The production cost and selling price of each product can be represented in a system of equations, and solving this system can help determine the optimal production levels to maximize profit. In engineering, systems of equations are used extensively in circuit analysis, structural analysis, and control systems. Electrical circuits, for example, can be modeled using Kirchhoff's laws, which result in a system of linear equations. Solving these equations provides the current flowing through different parts of the circuit. In structural analysis, systems of equations are used to determine the forces acting on different parts of a structure, ensuring its stability and safety. Computer science also utilizes systems of equations in various applications, such as computer graphics, cryptography, and network analysis. In computer graphics, systems of equations are used to perform transformations, such as rotations and scaling, on 3D objects. Cryptography relies on complex mathematical systems, including systems of equations, to encrypt and decrypt data securely. In network analysis, systems of equations can be used to model and analyze network traffic flow, optimizing network performance. Beyond these specific fields, systems of equations are also applicable in everyday situations. For example, when planning a budget, one might use a system of equations to allocate funds to different categories while staying within a certain total budget. Similarly, in cooking, recipes often involve ratios that can be expressed as a system of equations, helping to scale the recipe up or down. These examples illustrate the broad applicability of systems of equations in real-world contexts. Mastering the techniques for solving systems of equations provides a valuable tool for tackling a wide range of practical problems.
Conclusion
In conclusion, solving systems of equations is a fundamental skill in algebra with widespread applications. This comprehensive guide has walked through solving the specific system x + y = 6 } and { x + 3y = -2 } using three primary methods and { x + 3y = -2 } is x = 10 and y = -4. This solution was verified by substituting these values back into the original equations, ensuring accuracy. Moreover, we explored the real-world applications of systems of equations, highlighting their relevance in economics, engineering, computer science, and everyday scenarios. These applications demonstrate the practical value of mastering this algebraic concept. By understanding and applying these methods, you can confidently solve a wide range of systems of equations. Whether you are a student, a professional, or simply someone interested in mathematics, the ability to solve systems of equations is a valuable asset. This guide has provided a solid foundation for further exploration and application of this important topic. Remember, practice is key to mastering any mathematical skill, so continue to solve various systems of equations to enhance your proficiency and confidence. With a strong grasp of these methods, you will be well-equipped to tackle more complex mathematical problems and real-world challenges.
