Solving System Of Equations Solution Verification And Steps
What is the step-by-step solution to the system of equations 2x - 3y = -1 and 11x - 9y = -13, and how is the solution (-2, -1) verified?
In the realm of mathematics, solving systems of equations is a fundamental skill. These systems represent a set of two or more equations with the same variables, and the solution represents the values of those variables that satisfy all equations simultaneously. In this article, we will delve into solving a specific system of equations, demonstrating a step-by-step approach to arrive at the solution. We will also explore how manipulating the equations can simplify the process and lead to the desired result. Understanding these techniques is crucial for various mathematical and real-world applications, from physics and engineering to economics and computer science. This article will equip you with the knowledge to confidently tackle similar problems and deepen your understanding of linear algebra.
The given system of equations is:
2x - 3y = -1
11x - 9y = -13
The solution to this system is stated to be (-2, -1), meaning that x = -2 and y = -1. Let's verify this solution and explore the method used to obtain it. We will also analyze the statement that multiplying the first equation by -3 and summing it with the second equation results in 5x = -10. This article will provide a comprehensive understanding of the steps involved in solving this system and the underlying principles of linear equations.
To verify that the solution (-2, -1) is correct, we substitute x = -2 and y = -1 into both equations:
For the first equation, 2x - 3y = -1:
2(-2) - 3(-1) = -4 + 3 = -1
The first equation is satisfied.
For the second equation, 11x - 9y = -13:
11(-2) - 9(-1) = -22 + 9 = -13
The second equation is also satisfied. Therefore, the solution (-2, -1) is indeed correct. This process of verification is crucial in ensuring the accuracy of our solution. By substituting the values back into the original equations, we can confirm that they hold true. This step helps prevent errors and builds confidence in our answer. In more complex systems, this verification step becomes even more critical, as it can quickly identify any mistakes made during the solving process. Furthermore, understanding how to verify solutions reinforces the fundamental concept of what it means for a point to be a solution to a system of equations – it must satisfy all equations simultaneously.
The problem states that multiplying the first equation by -3 and adding it to the second equation results in 5x = -10. Let's perform this operation to see if it holds true:
Multiply the first equation (2x - 3y = -1) by -3:
-3(2x - 3y) = -3(-1)
-6x + 9y = 3
Now, add this modified equation to the second equation (11x - 9y = -13):
(-6x + 9y) + (11x - 9y) = 3 + (-13)
5x = -10
This confirms the statement that the sum of the two equations after the manipulation is 5x = -10. This manipulation technique is a key element in solving systems of equations. By multiplying one or both equations by a constant and then adding or subtracting them, we can eliminate one variable. This simplifies the system, allowing us to solve for the remaining variable. In this case, the y variable was eliminated, leaving us with a single equation in terms of x. This method, often referred to as the elimination method, is a powerful tool in linear algebra. Understanding how to strategically manipulate equations is crucial for efficiently solving complex systems. The ability to identify the right multipliers to eliminate variables is a skill that comes with practice and a strong understanding of algebraic principles.
From the equation 5x = -10, we can solve for x:
5x = -10
x = -10 / 5
x = -2
This confirms that x = -2 is part of the solution. Solving for x in the simplified equation is a straightforward process. By isolating x on one side of the equation, we can directly determine its value. This step is a critical component of the elimination method, as it provides the numerical value for one of the variables in the system. Once we have the value of x, we can substitute it back into one of the original equations to solve for y. This iterative process of solving for one variable and then using its value to find the other is a common strategy in solving systems of equations. The clarity and simplicity of this step highlight the power of algebraic manipulation in reducing complex problems to manageable steps.
Now that we have x = -2, we can substitute it into either of the original equations to solve for y. Let's use the first equation, 2x - 3y = -1:
2(-2) - 3y = -1
-4 - 3y = -1
-3y = 3
y = -1
This confirms that y = -1 is the other part of the solution. Substituting the value of x back into one of the original equations is the final step in solving for y. This process allows us to leverage the value we found for x to determine the corresponding value of y. The choice of which equation to use is often a matter of convenience, as both equations should yield the same result. In this case, using the first equation provided a direct path to solving for y. This step underscores the interconnectedness of the variables in a system of equations. The value of one variable directly influences the value of the other, and by systematically solving for each, we arrive at the complete solution to the system. This reinforces the idea that a solution to a system of equations is a set of values that satisfy all equations simultaneously.
In conclusion, the solution to the system of equations:
2x - 3y = -1
11x - 9y = -13
is indeed (-2, -1). We verified this solution by substituting x = -2 and y = -1 into both equations and confirming that they hold true. We also demonstrated that multiplying the first equation by -3 and adding it to the second equation results in 5x = -10, which allowed us to solve for x. Finally, we substituted x = -2 back into the first equation to solve for y. This step-by-step approach illustrates the process of solving systems of linear equations using the elimination method. The ability to manipulate equations, eliminate variables, and solve for the unknowns is a fundamental skill in mathematics. This article has provided a clear and concise explanation of these techniques, equipping you with the knowledge to tackle similar problems with confidence. The principles discussed here extend to more complex systems of equations and are essential for various applications in science, engineering, and other fields. Understanding these concepts not only enhances your problem-solving abilities but also provides a solid foundation for further studies in mathematics and related disciplines.
Solving System of Equations Solution Verification and Steps