Solving System Of Linear Equations A And B
-x - 2y = 7
5x - 6y = -3
Solution: (-3, -2)
-x - 2y = 7
Choose the correct option that explains what steps were followed to obtain System B from System A.
Solving System of Linear Equations: A Detailed Guide
In the realm of mathematics, solving systems of linear equations is a fundamental skill. These systems, which involve two or more equations with the same variables, appear in various real-world applications, from engineering and economics to computer science and data analysis. Understanding the techniques to solve these systems is crucial for problem-solving and decision-making in many fields. This article will delve into the intricacies of solving a system of linear equations, specifically focusing on the system labeled as System A and how it relates to System B.
Understanding System A
System A is presented as follows:
-x - 2y = 7
5x - 6y = -3
This system consists of two linear equations, each containing two variables, x and y. A solution to this system is a pair of values for x and y that satisfies both equations simultaneously. We are given that the solution to System A is (-3, -2). This means that when we substitute x = -3 and y = -2 into both equations, the equations hold true. To verify this, let's substitute these values into the equations:
For the first equation:
-(-3) - 2(-2) = 3 + 4 = 7
For the second equation:
5(-3) - 6(-2) = -15 + 12 = -3
As we can see, the values x = -3 and y = -2 satisfy both equations, confirming that (-3, -2) is indeed the solution to System A. This verification step is crucial in ensuring the accuracy of the solution.
System B and the Transformation Process
System B is presented as:
-x - 2y = 7
Notice that System B only contains one equation, which is the first equation from System A. The question posed is: What steps were followed to obtain System B from System A? This question leads us to consider the various operations that can be performed on a system of equations without changing its solution set. These operations are essential for solving systems of equations using methods like elimination and substitution.
To understand how System B was obtained, we need to recognize that the second equation from System A is missing. This implies that some operation or process was applied to System A that effectively eliminated the second equation. Common operations that can be performed on a system of equations include:
- Substitution: Solving one equation for one variable and substituting that expression into the other equation.
- Elimination: Adding or subtracting multiples of the equations to eliminate one variable.
- Scalar Multiplication: Multiplying an equation by a constant.
- Row Operations: These are similar to elimination but are often used in the context of matrices.
In this specific case, since the second equation is entirely absent in System B, it's likely that the focus is on understanding which equation remains unchanged rather than how an equation was eliminated. The important observation here is that the first equation, -x - 2y = 7, is present in both System A and System B. This suggests that no operation was performed on this particular equation. The question then becomes: what happened to the second equation?
Methods to Solve Systems of Equations
To fully grasp the transformation from System A to System B, let's briefly discuss the common methods used to solve systems of linear equations. These methods will help us understand the possible operations that could have been applied to the original system.
1. Substitution Method:
The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation. This method reduces the system to a single equation with one variable, which can then be easily solved. Once the value of one variable is found, it can be substituted back into one of the original equations to find the value of the other variable.
For example, in System A, we could solve the first equation for x:
-x = 2y + 7
x = -2y - 7
Then, substitute this expression for x into the second equation:
5(-2y - 7) - 6y = -3
-10y - 35 - 6y = -3
-16y = 32
y = -2
Substituting y = -2 back into the expression for x:
x = -2(-2) - 7
x = 4 - 7
x = -3
Thus, the solution is x = -3 and y = -2.
2. Elimination Method:
The elimination method, also known as the addition method, involves manipulating the equations so that the coefficients of one variable are opposites. Then, the equations are added together, eliminating one variable and resulting in a single equation with one variable. This equation can be solved, and the value can be substituted back into one of the original equations to find the value of the other variable.
For System A, we can multiply the first equation by 5 to make the coefficients of x opposites:
5(-x - 2y) = 5(7)
-5x - 10y = 35
Now, add this modified equation to the second equation:
(-5x - 10y) + (5x - 6y) = 35 + (-3)
-16y = 32
y = -2
Substitute y = -2 back into the first equation:
-x - 2(-2) = 7
-x + 4 = 7
-x = 3
x = -3
Again, the solution is x = -3 and y = -2.
3. Matrix Method:
Systems of linear equations can also be solved using matrices. This method is particularly useful for larger systems with more variables. The system is represented as a matrix equation, and techniques like Gaussian elimination or matrix inversion are used to find the solution.
For System A, the matrix representation is:
| -1 -2 | | x | = | 7 |
| 5 -6 | | y | | -3 |
Solving this matrix equation involves finding the inverse of the coefficient matrix and multiplying it by the constant matrix. This method provides a systematic approach to solving complex systems of equations.
Analyzing the Transformation to System B
Given the various methods to solve systems of equations, we can now focus on what might have led to System B. The key observation is that only the first equation from System A is present in System B. This suggests that the second equation was either intentionally removed or became redundant in some way. Redundancy can occur if one equation is a linear combination of the other, meaning it provides no new information about the solution set.
However, in this case, the second equation in System A (5x - 6y = -3) is not a simple multiple of the first equation (-x - 2y = 7). Therefore, it is unlikely that the second equation became redundant through a simple scalar multiplication. It is more probable that the problem is designed to test understanding of equation manipulation rather than a complex mathematical operation.
Identifying the Correct Explanation
The most accurate explanation for the transformation from System A to System B is that the second equation was omitted or disregarded. This could be due to several reasons, such as the problem statement explicitly asking to consider only the first equation, or an exercise designed to assess understanding of individual equations within a system.
The solution (-3, -2) still satisfies the remaining equation in System B:
-(-3) - 2(-2) = 3 + 4 = 7
This confirms that the solution to the first equation in System A is consistent with System B, as expected. However, without the second equation, System B represents a single line in the xy-plane, and there are infinitely many solutions that satisfy it. The solution (-3, -2) is just one of them.
Conclusion
Understanding systems of linear equations is a crucial skill in mathematics, with applications spanning numerous fields. System A presents a typical system of two linear equations with a unique solution. System B, on the other hand, consists of only one equation from System A. The transformation from System A to System B most likely involves the intentional omission of the second equation, rather than a mathematical operation that eliminates it. This exercise highlights the importance of understanding the individual components of a system and how they contribute to the overall solution set. By mastering techniques like substitution, elimination, and matrix methods, one can effectively solve a wide range of problems involving linear equations.
In summary, the transition from System A to System B underscores the foundational principles of linear algebra and the significance of recognizing the individual roles of equations within a system. Whether it's for academic pursuits or real-world problem-solving, a firm grasp of these concepts is indispensable for anyone navigating the mathematical landscape.
