Solving Systems Of Equations By Elimination A Step By Step Guide
In the realm of algebra, solving systems of equations is a fundamental skill. One powerful method for tackling these systems is elimination, a technique that leverages the properties of equality to strategically remove variables and simplify the problem. This comprehensive guide will walk you through the process of solving a specific system of equations using elimination, providing detailed explanations and insights along the way.
The Given System of Equations
Let's consider the following system of equations:
8x + 7y = 39
4x - 14y = -68
Our goal is to find the values of x and y that simultaneously satisfy both equations. The elimination method provides a systematic approach to achieve this.
Step 1: Multiplying the First Equation for y-Term Elimination
The cornerstone of the elimination method lies in manipulating the equations to create opposite coefficients for one of the variables. In this case, we aim to eliminate the y term. Observe that the coefficient of y in the first equation is 7, while in the second equation, it's -14. To create opposite coefficients, we can multiply the first equation by 2. This will result in a y coefficient of 14, which is the opposite of -14.
Multiplying the entire first equation (8x + 7y = 39) by 2, we get:
2 * (8x + 7y) = 2 * 39
16x + 14y = 78
Now, our modified system of equations looks like this:
16x + 14y = 78
4x - 14y = -68
Notice that the y terms now have coefficients of 14 and -14, setting the stage for elimination.
Step 2: Adding the Equations to Eliminate y
The magic of elimination unfolds when we add the two equations together. By adding the left-hand sides and the right-hand sides separately, we can eliminate the y variable due to the opposite coefficients.
Adding the equations (16x + 14y = 78 and 4x - 14y = -68), we get:
(16x + 14y) + (4x - 14y) = 78 + (-68)
Combining like terms, we have:
16x + 4x + 14y - 14y = 10
20x = 10
The y terms have vanished, leaving us with a simple equation in terms of x. This is the essence of the elimination method – reducing the system to a single variable equation.
Step 3: Solving for x
With the equation 20x = 10, we can now easily solve for x. Dividing both sides by 20, we obtain:
x = 10 / 20
x = 1/2
Thus, we have found the value of x, which is 1/2.
Step 4: Substituting x to Solve for y
Now that we know x = 1/2, we can substitute this value into either of the original equations to solve for y. Let's choose the first original equation (8x + 7y = 39) for substitution. Substituting x = 1/2, we get:
8 * (1/2) + 7y = 39
4 + 7y = 39
Subtracting 4 from both sides, we have:
7y = 35
Dividing both sides by 7, we find:
y = 5
Therefore, the value of y is 5.
Step 5: Verifying the Solution
To ensure the correctness of our solution, it's crucial to verify that the values of x and y satisfy both original equations. Let's substitute x = 1/2 and y = 5 into both equations.
For the first equation (8x + 7y = 39):
8 * (1/2) + 7 * 5 = 39
4 + 35 = 39
39 = 39
The first equation holds true.
For the second equation (4x - 14y = -68):
4 * (1/2) - 14 * 5 = -68
2 - 70 = -68
-68 = -68
The second equation also holds true.
Since the values x = 1/2 and y = 5 satisfy both equations, we have successfully solved the system of equations.
Solution Summary
The solution to the system of equations is:
x = 1/2
y = 5
This can also be represented as the ordered pair (1/2, 5).
Conclusion
The elimination method provides a robust and systematic approach to solving systems of equations. By strategically manipulating equations to eliminate variables, we can simplify the problem and arrive at the solution. This step-by-step guide has demonstrated the application of the elimination method to a specific system of equations, highlighting the key principles and techniques involved. Mastering the elimination method empowers you to tackle a wide range of algebraic problems with confidence.
Remember, practice is key to proficiency. The more you work with systems of equations and the elimination method, the more comfortable and adept you will become at solving them. Embrace the challenge, and you'll unlock a powerful tool in your mathematical arsenal. Understanding systems of equations and the elimination method is crucial for various applications in mathematics, science, and engineering. By mastering this technique, you gain a valuable problem-solving skill that extends far beyond the classroom. The ability to eliminate variables and simplify complex equations is a cornerstone of mathematical reasoning, and it will serve you well in your academic and professional pursuits. Furthermore, the elimination method is not just a mathematical tool; it's a testament to the power of logical thinking and strategic manipulation. The process of identifying the right multipliers, adding equations, and solving for variables hones your analytical skills and enhances your ability to approach problems systematically. So, dive into the world of systems of equations and elimination, and discover the elegance and efficiency of this powerful technique. This method is not only effective but also demonstrates the beauty of mathematical problem-solving. The satisfaction of successfully eliminating variables and arriving at a solution is a rewarding experience that reinforces your understanding of algebraic principles. Continue to explore the nuances of the elimination method, and you'll find that it becomes an indispensable tool in your mathematical toolkit. The concepts of linear equations and their solutions are fundamental to many areas of mathematics and its applications. By grasping the core principles of the elimination method, you're laying a solid foundation for more advanced mathematical concepts. The elimination method is a classic technique that has stood the test of time, and it remains a valuable tool for solving systems of equations in various contexts. Whether you're a student, a teacher, or a professional, mastering the elimination method will undoubtedly enhance your mathematical capabilities. Embrace the opportunity to learn and practice this method, and you'll reap the benefits in your academic and professional endeavors. The journey of learning mathematics is filled with challenges and rewards, and mastering the elimination method is a significant milestone along the way. So, take the time to understand the underlying principles, practice the steps involved, and celebrate your successes as you become proficient in this powerful problem-solving technique.
By understanding how to eliminate variables effectively, you can tackle more complex systems and real-world applications with greater confidence. This method not only provides a solution but also enhances your understanding of mathematical relationships and problem-solving strategies. The process of solving for x and y requires careful attention to detail and a systematic approach, which are valuable skills applicable in various fields beyond mathematics. The ability to verify the solution is a crucial step in ensuring accuracy and building confidence in your problem-solving abilities. This practice reinforces the importance of checking your work and understanding the underlying principles of the mathematical operations involved.
