Henrique's Linear Equations Error Analysis And Corrected Solution

In this article, we will meticulously analyze Henrique's attempt to solve a system of linear equations using the linear combination method. We will pinpoint the exact location of the error in his work and provide a step-by-step solution to correctly solve the system. Furthermore, we will delve into the intricacies of the linear combination method, offering a comprehensive guide for students and enthusiasts alike. Understanding the nuances of this method is crucial for solving various mathematical problems, especially in algebra and calculus. Our goal is to not only correct Henrique's mistake but also to empower you with the knowledge and skills to confidently tackle similar problems. This article serves as a valuable resource for students learning about systems of linear equations, teachers looking for examples to illustrate common errors, and anyone interested in honing their mathematical problem-solving abilities. The principles discussed here extend beyond the specific problem at hand, offering insights applicable to a broader range of algebraic challenges.

Problem Statement

Henrique attempted to solve the following system of linear equations using the linear combination method:

1. $\begin{array}{l} 3(4 x-7 y=28) -12 x-21 y=84 \\ -2(6 x-5 y=31) -12 x+10 y=-62 \end{array}$

2. $\begin{aligned} 12Discussion category : \\ mathematics \end{aligned}$

Our objective is to identify the error in Henrique's approach and provide a corrected solution.

Identifying the Error

Upon examining Henrique's work, the error lies in the initial multiplication and distribution step. Specifically, let's break down each equation: The first equation was correctly multiplied by 3 on both sides, resulting in 12x - 21y = 84. However, the second equation presents an issue. When multiplying the equation 6x - 5y = 31 by -2, the result should be -12x + 10y = -62. Henrique correctly obtained -12x + 10y = -62. The next step in the linear combination method involves adding the two modified equations together. This is where a critical mistake can easily occur if the signs are not carefully managed. By adding the equations, the 'x' terms should cancel out, leaving an equation in terms of 'y'. The subsequent steps depend heavily on the accurate execution of this addition. Overlooking a negative sign or miscalculating the sum of the constants can lead to an incorrect solution. Understanding the principles behind the linear combination method helps in identifying and rectifying such errors, making the solution process smoother and more reliable. Therefore, careful attention to detail is essential when applying this method, especially when dealing with negative coefficients and constants.

Correcting the Solution: Step-by-Step Guide

To correctly solve the system of equations, let's retrace the steps with careful attention to detail. The first step involves multiplying each equation by a constant so that the coefficients of either x or y are opposites. As Henrique did, we can multiply the first equation by 3 and the second equation by -2:

• Equation 1: 3 * (4x - 7y) = 3 * 28 => 12x - 21y = 84
• Equation 2: -2 * (6x - 5y) = -2 * 31 => -12x + 10y = -62

The next step is to add the two resulting equations together. This eliminates the x variable:

(12x - 21y) + (-12x + 10y) = 84 + (-62)

Simplifying the equation:

-11y = 22

Now, solve for y by dividing both sides by -11:

y = -2

Having found the value of y, we can substitute it back into either of the original equations to solve for x. Let's use the first original equation:

4x - 7y = 28 4x - 7(-2) = 28 4x + 14 = 28 4x = 14 x = 14/4 = 7/2

Therefore, the correct solution to the system of equations is x = 7/2 and y = -2. This methodical approach, which includes careful multiplication, addition, and substitution, ensures accurate results when solving systems of linear equations. It's crucial to double-check each step to avoid common mistakes, particularly when dealing with negative signs and fractions. This step-by-step process not only leads to the correct answer but also builds a solid understanding of the underlying algebraic principles.

Understanding the Linear Combination Method

The linear combination method, also known as the elimination method, is a powerful technique for solving systems of linear equations. It relies on the principle of adding or subtracting multiples of equations to eliminate one variable, thereby simplifying the system. The core idea is to manipulate the equations in such a way that when they are added together, one of the variables cancels out, leaving a single equation with a single variable. This resulting equation can then be easily solved. The value obtained is then substituted back into one of the original equations to find the value of the other variable. The beauty of this method lies in its systematic approach, which makes it less prone to errors compared to other methods like substitution or graphing, especially when dealing with complex coefficients or fractions. Understanding the fundamental principles of the linear combination method is crucial for solving systems of equations efficiently and accurately. This method is not only a cornerstone of algebra but also has wide-ranging applications in higher mathematics and various fields of science and engineering, where systems of equations frequently arise.

Steps Involved

The linear combination method involves a series of well-defined steps: The first step is to align the equations, ensuring that like terms (terms with the same variable) are in the same column. This makes the subsequent addition or subtraction process more straightforward. Next, examine the coefficients of the variables you wish to eliminate. If necessary, multiply one or both equations by a constant so that the coefficients of one variable are opposites or equal. This is the crucial step that sets up the elimination. Then, add or subtract the equations. If the coefficients were made opposites, you add the equations; if they were made equal, you subtract. This action should eliminate one variable, leaving you with a single equation in one variable. Solve the resulting equation for the remaining variable. Once you have the value of one variable, substitute it back into either of the original equations to solve for the other variable. Finally, it's always a good practice to check your solution by substituting both values into both original equations to ensure they hold true. This step verifies that your solution is correct and satisfies the system of equations. Understanding and meticulously following these steps is key to successfully applying the linear combination method.

When to Use the Linear Combination Method

The linear combination method is particularly effective when the coefficients of one variable in the system of equations are either the same or opposites, or when they can easily be made so by multiplying the equations by constants. This makes the elimination step straightforward and efficient. Situations where the equations are already aligned with like terms stacked vertically also favor this method. While the linear combination method can be used for any system of linear equations, it may be more advantageous than other methods, such as substitution, when dealing with equations where variables have coefficients other than 1. For instance, in equations like 3x + 2y = 7 and 5x - 2y = 1, the linear combination method allows for direct elimination of y by simply adding the equations. In contrast, the substitution method would require more initial algebraic manipulation to isolate a variable. Therefore, recognizing the structure of the equations and choosing the most appropriate method can significantly streamline the problem-solving process. The linear combination method shines in scenarios where the equations are set up for easy elimination, making it a valuable tool in your mathematical arsenal.

Common Mistakes to Avoid

When employing the linear combination method, several common pitfalls can lead to incorrect solutions. One of the most frequent errors is the misapplication of the distributive property. When multiplying an equation by a constant, it's crucial to multiply every term in the equation, including the constant term on the other side of the equals sign. Forgetting to distribute the constant to all terms can drastically alter the equation and lead to a wrong answer. Another common mistake arises from sign errors, especially when adding or subtracting equations. Carelessly handling negative signs can change the outcome of the elimination process and result in an incorrect solution. It’s imperative to double-check the signs of each term when combining equations. A third pitfall is incorrectly substituting the value of one variable back into the original equations to solve for the other variable. If the substitution is performed incorrectly, the resulting value will be wrong. Always ensure you are substituting the value into the correct equation and performing the calculations accurately. Finally, failing to check the solution is a common oversight. After finding the values of the variables, substitute them back into both original equations to verify that they satisfy the system. This simple step can catch many errors and ensure the accuracy of your solution. Avoiding these common mistakes through careful attention to detail and systematic execution of the steps will significantly improve your success in solving systems of linear equations using the linear combination method.

Conclusion

In conclusion, Henrique's attempt to solve the system of linear equations highlights the importance of precision and attention to detail when applying the linear combination method. By identifying the error in the initial multiplication step and providing a corrected, step-by-step solution, we have demonstrated the correct application of this powerful algebraic technique. Furthermore, our comprehensive guide to the linear combination method, including the steps involved, when to use it, and common mistakes to avoid, equips readers with the knowledge and skills necessary to confidently solve similar problems. Mastering this method is crucial for success in algebra and beyond, as systems of linear equations arise in various mathematical and scientific contexts. This article serves as a valuable resource for students, educators, and anyone seeking to enhance their problem-solving abilities in mathematics. Remember, practice and careful attention to detail are key to mastering the linear combination method and achieving accurate solutions. As you continue to practice, you'll become more adept at identifying the most efficient way to solve a system of equations and avoid common errors, making this method a valuable asset in your mathematical toolkit.