Sylvie's System Of Equations Solution Analysis And Correction

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In the realm of mathematics, solving systems of equations is a fundamental skill with wide-ranging applications. This article delves into a specific example where Sylvie tackles a system of two linear equations. Before we dissect Sylvie's approach, let's clarify what a system of equations entails. A system of equations is a set of two or more equations containing the same variables. The goal is to find the values of these variables that satisfy all equations simultaneously. Graphically, the solution to a system of two equations represents the point(s) where the lines or curves intersect. There are several methods to solve systems of equations, including substitution, elimination, and graphing. Each method has its strengths and weaknesses, and the choice of method often depends on the specific structure of the equations. In this case, Sylvie employs the substitution method, a technique particularly well-suited when one equation is already solved for one variable in terms of the other. By substituting one expression into another equation, we effectively reduce the system to a single equation with a single variable, making it easier to solve. This problem is very useful for applying math skills. It allows us to understand how to find where two lines meet, which is a key concept in algebra and can be used in real-world situations like planning routes or managing resources. Sylvie's method of solving this problem gives us a clear way to approach these kinds of math challenges, showing us how to break them down step by step to find the answers. This skill is important not just for math class but also for using math in everyday life and in other subjects.

Sylvie's Approach: A Step-by-Step Breakdown

Sylvie is presented with the following system of equations:

y = 5x + 2
y = 8 - x

Notice that both equations are already solved for y. This makes the substitution method a natural choice. Sylvie's initial step is to recognize that since both expressions are equal to y, they must also be equal to each other. This leads to the equation:

8 - x = 5x + 2

This equation now contains only one variable, x, which we can solve using algebraic manipulation. Sylvie's next step is to isolate the x terms on one side of the equation and the constant terms on the other. She adds x to both sides and subtracts 2 from both sides, resulting in:

8 = 6x + 2

6 = 6x

Finally, she divides both sides by 6 to solve for x:

x = 1

Now that Sylvie has found the value of x, she can substitute it back into either of the original equations to find the value of y. Sylvie chooses the second equation, y = 8 - x, and substitutes x = 1:

y = 8 - 1
y = 7

Thus, Sylvie finds that x = 1 and y = 7. This represents the solution to the system of equations, the point where the two lines intersect on a graph. To ensure the solution is correct, Sylvie performs a crucial step: verification. She substitutes both x = 1 and y = 7 into both original equations:

7 = 5(1) + 2
7 = 7  (True)

7 = 8 - 1
7 = 7  (True)

Since both equations hold true, Sylvie confidently concludes that the solution x = 1 and y = 7 is correct. This methodical approach highlights Sylvie's understanding of the substitution method and her attention to detail in verifying the solution. This method is a cornerstone of algebra, enabling us to tackle problems involving multiple variables and relationships. By carefully manipulating equations and substituting values, we can unravel complex systems and find precise solutions. Sylvie's work here is a prime example of how to effectively apply these techniques to achieve accurate results.

The Flaw in the Conclusion: Infinite Solutions?

However, the final statement in the provided solution is incorrect. Sylvie correctly finds the solution x = 1 and y = 7. The verification step confirms that this solution satisfies both equations. Therefore, there is one unique solution, not infinite solutions. The misconception of infinite solutions often arises when dealing with dependent systems of equations. A dependent system is one where the two equations represent the same line. In other words, one equation is a multiple of the other. In such cases, there are indeed infinitely many solutions because any point on the line satisfies both equations. However, in this case, the two equations represent distinct lines with different slopes (5 and -1). They intersect at a single point, which Sylvie correctly found. The conclusion that there are infinite solutions is a critical error that undermines the entire solution process. It highlights the importance of not just performing the calculations correctly but also interpreting the results accurately. It is crucial to distinguish between systems with unique solutions, no solutions, and infinite solutions. A unique solution occurs when the lines intersect at a single point, as in this case. No solution occurs when the lines are parallel and never intersect. Infinite solutions occur when the lines are the same. Understanding these distinctions is essential for mastering the concept of systems of equations and applying it effectively in various mathematical and real-world contexts. Sylvie's work up to the verification step is flawless, demonstrating a strong grasp of the substitution method. However, the final conclusion reveals a misunderstanding of the nature of solutions to systems of equations. This underscores the need for a comprehensive understanding of the underlying concepts, not just the mechanical steps of solving equations.

Identifying the Error: Why Not Infinite Solutions?

The error in Sylvie's conclusion stems from a misinterpretation of the solution she obtained. Sylvie correctly found x = 1 and y = 7 as the solution to the system. This means that the two lines represented by the equations intersect at the point (1, 7). The fact that she found a specific solution implies that the system is independent, meaning the equations represent distinct lines that intersect at a single point. A system has infinite solutions only if the two equations represent the same line. This would happen if one equation was a multiple of the other. For example, if the system were:

y = 2x + 4
2y = 4x + 8

The second equation is simply twice the first equation. These equations represent the same line, and any point on that line is a solution to the system. In this case, however, the equations y = 5x + 2 and y = 8 - x are clearly different. They have different slopes (5 and -1) and different y-intercepts (2 and 8). This means they will intersect at exactly one point, which Sylvie correctly calculated. The mistake likely arises from a confusion between different types of systems of equations. It's important to remember the following:

  • Independent systems: Have one unique solution (lines intersect at one point).
  • Inconsistent systems: Have no solution (lines are parallel).
  • Dependent systems: Have infinite solutions (lines are the same).

Sylvie's system falls into the first category – it's an independent system with one unique solution. Therefore, the conclusion of infinite solutions is incorrect. To prevent such errors, it's helpful to visualize the lines represented by the equations. If you graph the lines, you'll see that they intersect at a single point, confirming the unique solution. Additionally, after finding a solution, it's always wise to double-check the nature of the system to ensure the conclusion aligns with the results. In this case, recognizing that the lines have different slopes should have immediately ruled out the possibility of infinite solutions.

Conclusion: The Importance of Accurate Interpretation

In summary, Sylvie's solution demonstrates a strong understanding of the substitution method for solving systems of equations. She correctly manipulates the equations, finds the values of x and y, and verifies the solution. However, the final conclusion that the system has infinite solutions is incorrect. The system has one unique solution: x = 1, y = 7. This example underscores the importance of not only performing calculations accurately but also interpreting the results correctly. It highlights the need for a comprehensive understanding of the concepts underlying mathematical procedures. In this case, the misunderstanding lies in the distinction between independent, inconsistent, and dependent systems of equations. To avoid such errors, it's crucial to:

  • Understand the different types of systems of equations.
  • Visualize the lines represented by the equations.
  • Double-check the conclusion against the calculated solution and the nature of the system.

Sylvie's work provides a valuable learning opportunity. It shows that even with correct calculations, a misinterpretation can lead to an incorrect conclusion. By focusing on conceptual understanding and careful analysis, we can avoid such errors and strengthen our problem-solving skills in mathematics. The ability to solve systems of equations is not just a theoretical exercise; it's a practical skill with applications in various fields, including engineering, economics, and computer science. Mastering this skill requires both procedural fluency and conceptual understanding. This example serves as a reminder that both aspects are equally important for success in mathematics and beyond. Therefore, always check that the solution makes sense in context and aligns with the underlying mathematical principles. In doing so, we can solve the equation accurately and interpret the results with the best ability.