Solving Linear Equations By Linear Combination Analyzing Henry's Approach

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Introduction

In the realm of mathematics, particularly in linear algebra, solving systems of linear equations is a fundamental skill. These systems, which represent relationships between variables, can model a wide array of real-world phenomena, from circuit analysis in electrical engineering to resource allocation in economics. Among the various methods available for solving these systems, the linear combination method stands out for its elegance and efficiency. This article delves into the intricacies of this method, specifically addressing a scenario involving the equations 3x - 2y = 4 and 9x - 6y = 12. We will analyze a specific approach suggested by "Henry" and discuss its validity and implications.

The linear combination method, also known as the elimination method, hinges on the principle of manipulating equations to eliminate one variable, thereby simplifying the system. This manipulation typically involves multiplying one or both equations by constants and then adding or subtracting the equations. The key is to choose multipliers that will result in the coefficients of one variable becoming opposites, allowing them to cancel out upon addition. This process reduces the system to a single equation with a single variable, which can then be easily solved. The solution for this variable is then substituted back into one of the original equations to find the value of the other variable.

In the given system, we have two linear equations: 3x - 2y = 4 and 9x - 6y = 12. Henry proposes a specific strategy: multiply the first equation by -3 and then add the two equations together. This approach aims to eliminate the x-terms. To understand why this strategy might be considered, let's examine the coefficients of x in both equations. The first equation has a coefficient of 3, while the second equation has a coefficient of 9. Multiplying the first equation by -3 would result in a new coefficient of -9 for x. Adding this modified equation to the second equation, which has a coefficient of 9 for x, should indeed eliminate the x-term, as -9 + 9 = 0. However, the consequences of this action for the y-terms and the constants on the right-hand side need further scrutiny. We must also consider the broader implications of this operation for the solution set of the system.

Henry's Approach: A Detailed Examination

Let's meticulously examine Henry's approach to solving the system of equations:

  1. Multiply the first equation by -3: The first equation is 3x - 2y = 4. Multiplying both sides of this equation by -3 yields: -3(3x - 2y) = -3(4) -9x + 6y = -12

  2. Add the modified equation to the second equation: The modified first equation is -9x + 6y = -12, and the second equation is 9x - 6y = 12. Adding these two equations together, we get: (-9x + 6y) + (9x - 6y) = -12 + 12 -9x + 9x + 6y - 6y = 0 0 = 0

The result, 0 = 0, is a true statement, but it doesn't provide specific values for x and y. This outcome indicates that the two original equations are not independent; they are essentially the same equation represented in different forms. In other words, one equation is a multiple of the other. This type of system is known as a dependent system, and it has infinitely many solutions. To illustrate this, notice that if you multiply the first equation (3x - 2y = 4) by 3, you obtain the second equation (9x - 6y = 12). This redundancy is the reason why the x-terms and the y-terms both canceled out, leaving us with an identity (0 = 0) instead of a solution.

Implications of a Dependent System

When a system of linear equations is dependent, it means the equations represent the same line (in the case of two variables) or the same plane (in the case of three variables), or a higher-dimensional equivalent. Consequently, there isn't a unique point of intersection that satisfies both equations simultaneously; rather, every point on the line (or plane, etc.) is a solution. This infinite solution set can be expressed in parametric form. To do this, we can solve one of the equations for one variable in terms of the other.

Let's take the first equation, 3x - 2y = 4, and solve for x:

3x = 2y + 4 x = (2y + 4) / 3

Now we can express the solution set as a set of ordered pairs (x, y) where x = (2y + 4) / 3 and y can be any real number. This representation shows that for every value we choose for y, we can calculate a corresponding value for x that satisfies both equations. For example, if y = 1, then x = (2(1) + 4) / 3 = 2, so (2, 1) is a solution. If y = 4, then x = (2(4) + 4) / 3 = 4, so (4, 4) is another solution. We can generate infinitely many solutions in this manner.

Alternative Approaches and the Importance of Recognizing Dependent Systems

While Henry's approach correctly identified that multiplying the first equation by -3 and adding it to the second would eliminate x, the result (0 = 0) indicates that a different strategy or a different interpretation of the result is necessary to fully solve the system. It is crucial to recognize that the 0 = 0 outcome signifies a dependent system with infinitely many solutions. If the result had been a contradiction, such as 0 = 5, it would have indicated an inconsistent system with no solutions. Had the equations been independent, a unique solution for x and y would have been obtained.

An alternative approach to solving this system could involve solving one of the equations for one variable and then substituting that expression into the other equation. For instance, we could solve the first equation for y:

3x - 2y = 4 -2y = -3x + 4 y = (3x - 4) / 2

Then substitute this expression for y into the second equation:

9x - 6((3x - 4) / 2) = 12 9x - 3(3x - 4) = 12 9x - 9x + 12 = 12 12 = 12

Again, we arrive at a true statement, confirming the dependent nature of the system. This method, known as the substitution method, provides another route to identifying a dependent system. The graphical method is another powerful tool. If we were to graph the two equations, we would see that they overlap, visually demonstrating the infinite solutions.

The Significance of Understanding Linear Systems

The ability to solve systems of linear equations is fundamental in various fields. In engineering, these systems can model electrical circuits, structural mechanics, and control systems. In economics, they are used to analyze market equilibrium and resource allocation. Computer graphics and image processing rely heavily on linear algebra, including the solution of linear systems, for transformations and rendering. Furthermore, many numerical methods used in scientific computing, such as those for solving differential equations, rely on the efficient solution of large linear systems.

Understanding the different types of linear systems – independent, dependent, and inconsistent – is crucial for correctly interpreting the solutions and their implications. A unique solution indicates a well-defined relationship between the variables, while infinitely many solutions suggest a redundancy or an underdetermined situation. No solution points to a contradiction in the model, potentially requiring a revision of the underlying assumptions.

Conclusion

Henry's initial step in multiplying the first equation by -3 was a valid application of the linear combination method, but the subsequent result of 0 = 0 revealed that the system was dependent, possessing infinitely many solutions. This outcome highlights the importance of not only applying algebraic manipulations correctly but also interpreting the results within the context of the system's properties. The ability to recognize dependent and inconsistent systems is a critical skill in mathematics and its applications. While Henry's method didn't directly lead to a unique solution, it successfully illuminated the nature of the system, paving the way for a complete understanding of the solution set. Solving linear equations using methods like linear combination is a cornerstone of mathematical problem-solving, with applications spanning across numerous scientific and engineering disciplines. The deeper understanding of these methods and their implications provides a solid foundation for tackling more complex mathematical challenges.