Solving Linear Equations By Elimination Tao's Method
In the realm of mathematics, particularly in linear algebra, solving systems of equations is a fundamental skill. One powerful technique for achieving this is the elimination method. Let's delve into how Tao employs this method to solve a specific system of linear equations. This comprehensive guide will not only walk you through the solution step-by-step but also provide a deeper understanding of the underlying principles. Understanding how to solve systems of linear equations is crucial in various fields, including engineering, economics, and computer science. The elimination method, also known as the addition method, is particularly effective when the coefficients of one variable in the equations are opposites or can easily be made opposites through multiplication. This allows us to eliminate one variable by adding the equations together, resulting in a single equation with one unknown. This method is especially beneficial when dealing with larger systems of equations, as it provides a systematic way to reduce the complexity of the problem. By mastering the elimination method, you'll be equipped to tackle a wide range of mathematical challenges and real-world applications.
The Given System of Equations
Tao is confronted with the following system of two linear equations:
-2x + 4y = 16
2x + 2y = 8
Our goal is to find the ordered pair (x, y) that satisfies both equations simultaneously. This ordered pair represents the point where the two lines represented by these equations intersect on a coordinate plane. Solving systems of linear equations is a cornerstone of algebra and has practical applications in various fields, such as economics, engineering, and computer science. For example, in economics, these systems can model supply and demand relationships, while in engineering, they can be used to analyze circuits or structural systems. Therefore, mastering the techniques for solving these systems is crucial for both academic and professional success. The given system of equations presents a classic scenario where the elimination method shines. Notice how the coefficients of the x terms are opposites (-2 and 2). This sets the stage for a straightforward elimination process, where adding the equations will directly eliminate the x variable. This strategic observation is key to efficiently solving the system. By recognizing such opportunities, you can streamline your problem-solving approach and arrive at the solution more quickly.
Step 1: Eliminating 'x'
The beauty of this system lies in the fact that the coefficients of 'x' are already opposites (-2 and 2). This makes the elimination process incredibly straightforward. To eliminate 'x', we simply add the two equations together:
(-2x + 4y) + (2x + 2y) = 16 + 8
This simplifies to:
6y = 24
The elimination method works by strategically adding or subtracting equations to eliminate one variable, making it easier to solve for the remaining variable. In this case, the coefficients of 'x' were designed to cancel each other out when the equations were added, leaving us with a single equation in terms of 'y'. This is a common technique in solving systems of linear equations and highlights the power of algebraic manipulation. The goal is to transform the system into a simpler form that can be easily solved. By carefully choosing which operations to perform, we can systematically reduce the complexity of the problem and arrive at the solution efficiently. This step-by-step approach is crucial for maintaining accuracy and avoiding errors.
Step 2: Solving for 'y'
Now we have a simple equation with one variable:
6y = 24
To solve for 'y', we divide both sides of the equation by 6:
y = 24 / 6
y = 4
Thus, the value of 'y' is 4. Solving for a single variable is a fundamental skill in algebra. Once we have isolated the variable, we can determine its value by performing the inverse operation. In this case, since 'y' is multiplied by 6, we divide both sides of the equation by 6 to isolate 'y'. This straightforward process is essential for solving a wide range of mathematical problems. It's important to remember that whatever operation we perform on one side of the equation, we must also perform on the other side to maintain the equality. This principle is the foundation of algebraic manipulation and ensures that the equation remains balanced throughout the solving process. By mastering these basic algebraic techniques, you'll be well-equipped to tackle more complex equations and systems of equations.
Step 3: Substituting 'y' to Find 'x'
Now that we know y = 4, we can substitute this value into either of the original equations to solve for 'x'. Let's use the second equation:
2x + 2y = 8
Substitute y = 4:
2x + 2(4) = 8
2x + 8 = 8
Subtract 8 from both sides:
2x = 0
Divide by 2:
x = 0
So, the value of 'x' is 0. Substitution is a powerful technique for solving systems of equations. Once we have found the value of one variable, we can substitute it back into one of the original equations to solve for the other variable. This process allows us to leverage the information we've already obtained to make further progress towards the solution. The key is to choose an equation that is relatively simple to work with, minimizing the chances of making errors. In this case, the second equation (2x + 2y = 8) was a good choice because it had smaller coefficients, making the arithmetic easier. By carefully selecting the equation and performing the substitution correctly, we can efficiently determine the value of the remaining variable and complete the solution.
Step 4: The Solution
We have found that x = 0 and y = 4. Therefore, the solution to the system of equations is the ordered pair (0, 4). This ordered pair represents the point where the two lines represented by the original equations intersect on a coordinate plane. The solution to a system of equations is the set of values that satisfy all the equations simultaneously. In the case of linear equations, this solution often corresponds to the point of intersection of the lines. However, it's important to note that systems of equations can have one solution, no solutions (parallel lines), or infinitely many solutions (the same line). Understanding these different possibilities is crucial for interpreting the results and applying them in real-world contexts. The ordered pair (0, 4) is the unique solution to this system, meaning that it is the only point that lies on both lines.
The Answer
Comparing our solution (0, 4) to the given options, we find that it matches option A.
Therefore, the correct answer is:
A. (0, 4)
Choosing the correct answer from a set of options is a crucial skill in mathematics assessments. After solving the problem, it's essential to carefully compare your solution to the provided options to ensure accuracy. In this case, we systematically solved the system of equations using the elimination method and arrived at the ordered pair (0, 4). This solution perfectly matches option A, confirming that it is the correct answer. This final step reinforces the importance of attention to detail and the ability to translate a mathematical solution into a specific answer choice. By practicing this skill, you can improve your performance on tests and quizzes and build confidence in your problem-solving abilities. The process of verifying the solution against the options also serves as a check for any potential errors made during the solution process.
Conclusion
Tao successfully used the elimination method to solve the system of linear equations. The solution, the ordered pair (0, 4), represents the point where the two lines intersect. This example demonstrates the power and efficiency of the elimination method in solving systems of linear equations. The elimination method is a versatile tool for solving systems of linear equations, particularly when the coefficients of one variable are opposites or can be easily made opposites. This method provides a systematic approach for reducing the complexity of the system and finding the solution. By mastering this technique, you'll be well-equipped to tackle a wide range of mathematical problems and real-world applications. The key to success lies in understanding the underlying principles, practicing the steps, and carefully checking your work. With consistent effort, you can confidently solve systems of linear equations using the elimination method and apply this skill to various contexts. Remember to always double-check your calculations and verify your solution to ensure accuracy and build confidence in your problem-solving abilities.
