Solving 2x + 4y = 42 And 6x - 4y = 30 A Step-by-Step Guide
Introduction to Systems of Equations
In the realm of mathematics, systems of equations play a crucial role in solving real-world problems involving multiple variables and constraints. A system of equations is a set of two or more equations containing the same variables. The solutions to a system of equations are the values that, when substituted for the variables, make all the equations in the system true. Mastering the techniques to solve these systems is fundamental for various applications, ranging from simple algebraic problems to complex scientific and engineering calculations. This article delves into the process of solving a specific system of linear equations, providing a step-by-step guide that will enhance your understanding and problem-solving skills. The focus will be on the system:
2x + 4y = 42
6x - 4y = 30
We will explore various methods to find the values of x and y that satisfy both equations simultaneously. This detailed exploration aims to equip you with the knowledge and confidence to tackle similar problems effectively. Understanding these methods not only improves your mathematical proficiency but also enhances your analytical thinking, which is a valuable asset in numerous fields. Whether you are a student learning algebra or a professional needing to solve complex problems, this guide will serve as a comprehensive resource. The elegance and efficiency of these methods underscore the power of algebraic techniques in simplifying complex problems into manageable steps. By the end of this guide, you will have a solid understanding of how to approach and solve systems of linear equations, setting a strong foundation for more advanced mathematical concepts.
Methods to Solve Systems of Equations
There are several methods to solve systems of equations, each with its own advantages and best-use scenarios. We will primarily focus on the elimination method and the substitution method in this article, as they are particularly effective for solving the given system. However, it's beneficial to have an overview of other methods as well.
1. Elimination Method
The elimination method, also known as the addition or subtraction method, involves manipulating the equations so that when they are added or subtracted, one of the variables is eliminated. This reduces the system to a single equation with one variable, which can then be easily solved. Once one variable is found, it can be substituted back into one of the original equations to find the value of the other variable. This method is particularly useful when the coefficients of one of the variables are the same or can be easily made the same through multiplication. The beauty of the elimination method lies in its simplicity and directness, making it a favorite among mathematicians and students alike. Its systematic approach ensures that you can tackle complex systems with confidence. The key is to identify the most efficient way to eliminate a variable, which often involves looking for coefficients that are multiples of each other or have opposite signs. By mastering this method, you gain a powerful tool for solving a wide range of algebraic problems.
2. Substitution Method
The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This again reduces the system to a single equation with one variable. Once this variable is found, its value is substituted back into one of the original equations (or the expression derived earlier) to find the value of the other variable. This method is especially useful when one of the equations can be easily solved for one variable in terms of the other. The substitution method is a versatile technique that allows you to express one variable as a function of another, thereby simplifying the system. It's a fundamental concept in algebra and is widely used in higher-level mathematics. The elegance of this method lies in its ability to transform a system of equations into a single-variable equation, making it easier to solve. By understanding the substitution method, you not only enhance your problem-solving skills but also gain a deeper appreciation for the interconnectedness of algebraic expressions.
3. Graphical Method
The graphical method involves plotting the equations on a coordinate plane. The solution to the system is the point where the lines intersect. This method is particularly useful for visualizing the solutions and understanding the nature of the system (e.g., whether the system has a unique solution, no solution, or infinitely many solutions). While it might not be the most precise method for finding exact solutions, it provides a valuable visual representation of the system. The graphical method is an excellent tool for gaining intuition about the behavior of linear equations. It allows you to see the geometric interpretation of solutions, which can be incredibly helpful in understanding more complex systems. However, it's important to note that the accuracy of the solution depends on the precision of the graph, making it less suitable for situations requiring high precision.
4. Matrix Methods
Matrix methods, such as using the inverse of a matrix or Cramer's Rule, are more advanced techniques used for solving systems of linear equations, especially when dealing with larger systems. These methods leverage the power of linear algebra to efficiently find solutions. They are particularly useful in computational settings where systems of equations are solved programmatically. Matrix methods provide a systematic and efficient way to handle complex systems, making them indispensable in various scientific and engineering applications. These methods not only simplify the process of solving equations but also offer a powerful framework for analyzing the properties of the system, such as its consistency and uniqueness of solutions. Understanding matrix methods opens the door to advanced mathematical techniques and computational tools, essential for tackling real-world problems.
Solving the System 2x + 4y = 42 and 6x - 4y = 30
Now, let's apply these methods to solve the given system of equations:
2x + 4y = 42 (Equation 1)
6x - 4y = 30 (Equation 2)
1. Elimination Method in Detail
Looking at the equations, we notice that the coefficients of y are +4 and -4. This makes the elimination method an ideal choice. The y terms will cancel out when we add the two equations together. This direct cancellation simplifies the process and reduces the chance of errors. The elimination method, in this case, offers a straightforward path to the solution, making it an efficient and reliable approach. By focusing on the structure of the equations, we can quickly identify the most effective strategy for solving the system. This highlights the importance of pattern recognition and strategic thinking in mathematics.
Add Equation 1 and Equation 2:
(2x + 4y) + (6x - 4y) = 42 + 30
This simplifies to:
8x = 72
Now, solve for x:
x = 72 / 8
x = 9
Now that we have the value of x, substitute it back into one of the original equations to find y. Let's use Equation 1:
2(9) + 4y = 42
18 + 4y = 42
Subtract 18 from both sides:
4y = 42 - 18
4y = 24
Now, solve for y:
y = 24 / 4
y = 6
Thus, the solution to the system using the elimination method is x = 9 and y = 6. This solution satisfies both equations, confirming its correctness. The step-by-step process demonstrates the clarity and effectiveness of the elimination method. By systematically eliminating one variable, we transformed the system into a simple equation, making it easy to find the values of x and y. This approach underscores the importance of breaking down complex problems into manageable steps.
2. Substitution Method in Detail
To use the substitution method, we first need to solve one of the equations for one variable in terms of the other. Let's solve Equation 1 for x:
2x + 4y = 42
Subtract 4y from both sides:
2x = 42 - 4y
Divide both sides by 2:
x = 21 - 2y
Now, substitute this expression for x into Equation 2:
6(21 - 2y) - 4y = 30
Distribute the 6:
126 - 12y - 4y = 30
Combine like terms:
126 - 16y = 30
Subtract 126 from both sides:
-16y = 30 - 126
-16y = -96
Divide both sides by -16:
y = -96 / -16
y = 6
Now that we have the value of y, substitute it back into the expression for x:
x = 21 - 2(6)
x = 21 - 12
x = 9
Thus, the solution to the system using the substitution method is x = 9 and y = 6, which is the same solution we obtained using the elimination method. This consistency reinforces the accuracy of our calculations and demonstrates the versatility of these methods. The substitution method, while requiring a different set of steps, leads to the same correct answer, highlighting the robustness of algebraic techniques. By mastering both elimination and substitution, you equip yourself with powerful tools for solving a wide range of systems of equations.
Verification of the Solution
To ensure the accuracy of our solution, it's crucial to verify that the values x = 9 and y = 6 satisfy both equations in the original system. Verification is a critical step in problem-solving, as it helps catch any potential errors and builds confidence in the solution. This process not only confirms the correctness of the answer but also reinforces the understanding of the problem-solving process. By systematically checking the solution against the original equations, we can be certain that we have arrived at the correct answer.
Verification for Equation 1: 2x + 4y = 42
Substitute x = 9 and y = 6 into Equation 1:
2(9) + 4(6) = 42
18 + 24 = 42
42 = 42
The equation holds true. This confirms that our solution satisfies the first equation. The verification process provides a tangible confirmation that the values we found are indeed correct. It's a simple yet powerful step that ensures accuracy and prevents potential errors. This reinforces the importance of meticulousness in mathematics, where attention to detail can make a significant difference.
Verification for Equation 2: 6x - 4y = 30
Substitute x = 9 and y = 6 into Equation 2:
6(9) - 4(6) = 30
54 - 24 = 30
30 = 30
The equation holds true. This confirms that our solution also satisfies the second equation. With both equations verified, we can confidently conclude that x = 9 and y = 6 is the correct solution to the system. The verification process not only validates our answer but also reinforces our understanding of the system of equations. By ensuring that the solution works for all equations, we gain a deeper insight into the relationships between the variables.
Real-World Applications of Systems of Equations
Systems of equations are not just abstract mathematical concepts; they have numerous real-world applications across various fields. Understanding how to solve these systems allows us to model and solve complex problems in everyday life, science, engineering, economics, and more. The ability to translate real-world scenarios into mathematical models and solve them is a valuable skill. This section explores some of the key areas where systems of equations play a pivotal role. By recognizing these applications, we can appreciate the practical significance of mathematical concepts and their impact on our daily lives.
1. Engineering
In engineering, systems of equations are used to design structures, analyze circuits, and model dynamic systems. For example, in civil engineering, systems of equations can determine the forces and stresses in a bridge or building. In electrical engineering, they are used to analyze complex circuits and optimize their performance. These applications highlight the critical role of mathematics in ensuring the safety and efficiency of engineered systems. The precision and reliability offered by mathematical models are essential for designing structures that can withstand various conditions and operate effectively. By using systems of equations, engineers can make informed decisions and create innovative solutions to real-world problems.
2. Economics
Economics frequently uses systems of equations to model supply and demand, market equilibrium, and economic growth. These models help economists understand how different factors interact and influence economic outcomes. For instance, systems of equations can be used to determine the equilibrium price and quantity in a market, where supply equals demand. These insights are crucial for policymakers and businesses in making informed decisions about resource allocation and economic strategies. The ability to model economic phenomena using mathematical tools provides a powerful framework for analysis and prediction. By understanding the underlying equations, economists can develop strategies to promote economic stability and growth.
3. Physics
Physics relies heavily on systems of equations to describe the motion of objects, energy conservation, and various physical phenomena. For instance, projectile motion can be modeled using a system of equations that describes the object's horizontal and vertical positions over time. Similarly, circuit analysis in physics involves solving systems of equations to determine the currents and voltages in different parts of the circuit. These mathematical models are essential for understanding and predicting the behavior of physical systems. The precision offered by systems of equations allows physicists to make accurate predictions and develop new technologies based on fundamental principles.
4. Computer Science
In computer science, systems of equations are used in computer graphics, optimization problems, and algorithm design. For example, in computer graphics, systems of equations are used to transform and manipulate objects in 3D space. In optimization problems, systems of equations can help find the best solution among many possibilities. These applications highlight the interdisciplinary nature of mathematics and its importance in technological advancements. By leveraging mathematical tools, computer scientists can develop efficient algorithms and create innovative solutions for complex problems.
Conclusion
In conclusion, we have demonstrated how to solve the system of equations 2x + 4y = 42 and 6x - 4y = 30 using both the elimination and substitution methods. We found that the solution is x = 9 and y = 6. This solution was verified by substituting the values back into the original equations, ensuring its accuracy. The systematic approach to problem-solving, as demonstrated in this guide, is crucial for success in mathematics and related fields. By mastering these techniques, you can confidently tackle a wide range of problems and develop a deeper appreciation for the power of algebraic methods. Furthermore, we have highlighted the real-world applications of systems of equations, underscoring their importance in various fields such as engineering, economics, physics, and computer science. Understanding these applications helps to contextualize the mathematical concepts and appreciate their practical significance.
This comprehensive guide aims to provide you with a solid understanding of how to solve systems of equations and their relevance in real-world scenarios. Whether you are a student learning algebra or a professional applying mathematical tools, the techniques and insights discussed in this article will enhance your problem-solving skills and analytical thinking. By continuing to practice and explore different types of systems, you can further develop your mathematical proficiency and confidently tackle complex problems in any field.
