Solving For X And Y In The Equation (x+y, 2) = (10, X-y) A Step-by-Step Guide
Introduction to Solving Simultaneous Equations
In the realm of mathematics, solving simultaneous equations is a fundamental skill, particularly crucial in algebra and various advanced mathematical fields. Simultaneous equations, also known as systems of equations, involve finding the values of two or more variables that satisfy all equations within the system concurrently. These equations often represent real-world problems, making their solutions highly practical and applicable across diverse disciplines such as physics, engineering, economics, and computer science. This article delves into the method of solving for the variables x and y in a given equation, offering a step-by-step approach and detailed explanations to ensure a comprehensive understanding. This method involves equating corresponding components of ordered pairs, setting up a system of equations, and then employing algebraic techniques such as substitution or elimination to find the values of the variables. The ability to solve such equations is not only vital for academic pursuits but also for practical problem-solving in various professional contexts, highlighting its importance in mathematical education and real-world applications. Mastery of this skill empowers individuals to tackle complex problems with confidence and precision, underscoring its significance in the broader landscape of mathematical proficiency and its application in diverse fields.
Problem Statement: (x+y, 2) = (10, x-y)
The problem at hand presents us with an equation involving ordered pairs: (x+y, 2) = (10, x-y). To effectively solve for x and y, we must understand that two ordered pairs are equal if and only if their corresponding components are equal. This principle allows us to transform the single equation of ordered pairs into a system of two linear equations. By equating the first components, we establish the equation x + y = 10. Similarly, by equating the second components, we derive the equation 2 = x - y. We now have a system of two linear equations with two unknowns, which can be solved using several methods, including substitution, elimination, or matrix methods. The challenge lies in strategically applying these methods to efficiently determine the values of x and y that satisfy both equations simultaneously. This problem not only tests our algebraic skills but also our ability to translate a problem involving ordered pairs into a solvable system of equations, a common technique in various mathematical contexts. Solving this system will provide us with a unique solution for x and y that satisfies the original equation, demonstrating the practical application of solving simultaneous equations in a broader mathematical framework. The subsequent sections will explore the step-by-step solutions to effectively find these values.
Step-by-Step Solution
Step 1: Equating Corresponding Components
To begin solving the equation (x+y, 2) = (10, x-y), the first crucial step involves understanding the fundamental property of ordered pairs. Two ordered pairs are considered equal if and only if their corresponding components are equal. This principle allows us to break down the given equation into a system of two simpler equations. By equating the first components of the ordered pairs, we obtain our first equation: x + y = 10. This equation represents a linear relationship between x and y, where the sum of x and y is equal to 10. Next, we equate the second components of the ordered pairs, which gives us the second equation: 2 = x - y. This equation also represents a linear relationship between x and y, where the difference between x and y is equal to 2. Now, we have successfully transformed the original equation into a system of two linear equations:
- x + y = 10
- x - y = 2
This transformation is a critical step in solving for x and y, as it allows us to apply standard algebraic techniques for solving systems of equations. The system of equations we have obtained can be solved using various methods, such as substitution or elimination. The key is to choose a method that efficiently isolates one variable, allowing us to solve for its value, and then use that value to find the other variable. The following steps will demonstrate how to use the elimination method to solve this system of equations.
Step 2: Solving the System of Equations
Having established the system of equations:
- x + y = 10
- x - y = 2
we can now proceed to solve for x and y. The elimination method is particularly effective in this case due to the structure of the equations, where the y terms have opposite signs. By adding the two equations together, we can eliminate the y variable and solve for x directly. Adding equation (1) to equation (2), we get:
(x + y) + (x - y) = 10 + 2
This simplifies to:
2x = 12
Dividing both sides of the equation by 2, we find the value of x:
x = 6
Now that we have the value of x, we can substitute it back into either equation (1) or equation (2) to solve for y. Let's substitute x = 6 into equation (1):
6 + y = 10
Subtracting 6 from both sides, we get:
y = 4
Thus, we have found the values of x and y that satisfy the system of equations. The solution is x = 6 and y = 4. This method of elimination is a powerful tool for solving systems of equations, especially when the coefficients of one variable are opposites or can be easily made opposites through multiplication. The next step will verify this solution by substituting these values back into the original equations.
Step 3: Verifying the Solution
After finding the values x = 6 and y = 4, it is crucial to verify that these values indeed satisfy the original equation (x+y, 2) = (10, x-y). This verification step ensures that our solution is correct and that no algebraic errors were made during the solving process. To verify, we substitute x = 6 and y = 4 into both sides of the equation and check if the equality holds true. First, let's substitute the values into the left side of the equation (x+y, 2):
(6 + 4, 2) = (10, 2)
Next, we substitute the values into the right side of the equation (10, x-y):
(10, 6 - 4) = (10, 2)
Comparing the two sides, we see that (10, 2) = (10, 2), which confirms that our solution satisfies the original equation. This verification step is an essential part of the problem-solving process, as it provides confidence in the accuracy of the solution and highlights the importance of double-checking one's work in mathematics. By verifying our solution, we ensure that the values x = 6 and y = 4 are indeed the correct solution to the given equation. This comprehensive approach to problem-solving, including verification, is a key skill in mathematical proficiency.
Alternative Methods
While the elimination method provided an efficient way to solve the system of equations, alternative methods can also be used to find the values of x and y. One such method is the substitution method, which involves solving one equation for one variable and substituting that expression into the other equation. This section will explore how the substitution method can be applied to the same system of equations:
- x + y = 10
- x - y = 2
To use the substitution method, we can start by solving equation (1) for x:
x = 10 - y
Now, we substitute this expression for x into equation (2):
(10 - y) - y = 2
Simplifying the equation, we get:
10 - 2y = 2
Subtracting 10 from both sides:
-2y = -8
Dividing by -2:
y = 4
Now that we have the value of y, we can substitute it back into the expression for x:
x = 10 - 4
x = 6
Thus, the substitution method also yields the solution x = 6 and y = 4, confirming our previous result obtained using the elimination method. This demonstrates the versatility of algebraic techniques in solving systems of equations. Another alternative method could involve graphical solutions, where the equations are plotted on a coordinate plane, and the intersection point represents the solution. Each method offers a unique approach, and the choice of method often depends on the specific structure of the equations and personal preference. Understanding multiple methods enhances problem-solving skills and provides a broader perspective on mathematical concepts.
Conclusion
In conclusion, solving for x and y in the equation (x+y, 2) = (10, x-y) involves a systematic approach that begins with equating corresponding components to form a system of linear equations. This transformation is a fundamental step in solving such problems, allowing us to apply standard algebraic techniques. We then employed the elimination method, which proved to be an efficient way to find the values of x and y. By adding the two equations together, we eliminated the y variable, solved for x, and subsequently found the value of y through substitution. The solution obtained was x = 6 and y = 4. To ensure the accuracy of our solution, we performed a verification step, substituting the values back into the original equation and confirming that the equality holds true. Additionally, we explored an alternative method, the substitution method, which also yielded the same solution, highlighting the versatility of algebraic techniques in solving systems of equations. The ability to solve simultaneous equations is a critical skill in mathematics, with applications spanning various fields such as physics, engineering, economics, and computer science. Mastery of these techniques not only enhances problem-solving capabilities but also fosters a deeper understanding of mathematical concepts. This comprehensive approach to problem-solving, including verification and exploring alternative methods, is essential for mathematical proficiency and its application in diverse real-world scenarios. The principles and methods discussed in this article provide a solid foundation for tackling more complex mathematical problems and underscore the importance of algebraic skills in both academic and professional contexts.