Ernesto's Substitution Error Identifying Mistakes In Solving Equations

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In the realm of mathematics, solving systems of equations is a fundamental skill, crucial for various applications ranging from engineering to economics. Among the methods employed, substitution stands out as a powerful technique for finding solutions. However, the substitution method, while effective, is also prone to errors if not executed with precision. This article delves into a specific instance where Ernesto, a diligent student, attempts to solve a system of equations using substitution but encounters a misstep along the way. Our focus will be on dissecting Ernesto's work, pinpointing the exact step where the error occurs, and elucidating the correct approach to solving the system. By carefully examining each step of Ernesto's solution, we aim to not only identify the mistake but also to reinforce the underlying principles of the substitution method, ensuring a clear understanding for anyone tackling similar problems. This exploration serves as a valuable lesson in the importance of meticulousness and accuracy in mathematical problem-solving. The ability to identify and correct errors is just as important as the ability to execute the correct steps, and this analysis provides a practical example of how to develop that skill. Solving systems of equations is a core concept in algebra, and mastering this skill is essential for success in higher-level mathematics. Through this detailed analysis, we will reinforce the understanding of the substitution method and highlight common pitfalls to avoid. We will also demonstrate the importance of checking solutions to ensure their validity, a crucial step in any mathematical problem-solving process.

The Problem: A System of Equations

The problem at hand involves a system of two linear equations:

  1. x – y = 7
  2. 3x – 2y = 8

Ernesto's attempt to solve this system using substitution is presented step-by-step, allowing us to follow his thought process and identify any deviations from the correct method. Each step will be scrutinized to understand the transformations made and the underlying algebraic principles applied. The goal is to pinpoint the exact moment where the solution veers off course, providing a clear understanding of the error and its consequences. By breaking down the solution into individual steps, we can isolate the mistake and explain why it leads to an incorrect answer. This process not only helps in identifying the error but also in understanding the correct application of the substitution method. Recognizing and correcting errors is a vital part of learning mathematics, and this example provides a valuable opportunity to develop that skill. We will also discuss alternative approaches to solving the system of equations, such as the elimination method, to demonstrate the versatility of algebraic techniques.

Ernesto's Attempt: A Step-by-Step Analysis

Let's dissect Ernesto's attempt step by step:

  • Step 1: x = y + 7

    In the first step, Ernesto aims to isolate one variable in the first equation. By adding 'y' to both sides of the equation 'x – y = 7', he correctly derives 'x = y + 7'. This step is a standard initial move in the substitution method, where one variable is expressed in terms of the other. The goal is to create an expression that can be substituted into the second equation, effectively reducing the system to a single equation with one variable. This isolation step is crucial for the success of the substitution method, and Ernesto's execution here is flawless. It sets the stage for the subsequent steps, where the expression for 'x' will be used to eliminate 'x' from the second equation. The ability to manipulate equations in this way is a fundamental algebraic skill, and Ernesto demonstrates a clear understanding of this principle in this step. This initial step is a crucial foundation for the rest of the solution, and its correctness is essential for arriving at the correct answer. A mistake in this step would propagate through the rest of the solution, leading to an incorrect result.

  • Step 2: 3(y + 7) – 2y = 8

    Step 2 involves substituting the expression for 'x' obtained in Step 1 into the second equation. Replacing 'x' with '(y + 7)' in the equation '3x – 2y = 8' correctly yields '3(y + 7) – 2y = 8'. This is the core of the substitution method, where the isolated variable is used to eliminate one variable from the other equation. The substitution process is a direct application of the principle that if two expressions are equal, one can be substituted for the other without changing the validity of the equation. This step is crucial for transforming the system of two equations into a single equation with one unknown, which can then be solved using basic algebraic techniques. The accuracy of this substitution is paramount, as any error here will lead to an incorrect solution. Ernesto's execution of this step is accurate and demonstrates a clear understanding of the substitution method. The resulting equation, '3(y + 7) – 2y = 8', is now ready to be simplified and solved for 'y'. This step is a pivotal moment in the solution process, and its correct execution is a testament to Ernesto's understanding of the method.

  • Step 3: 3y + 21 – 2y = 8

    In Step 3, Ernesto expands the expression '3(y + 7)' by distributing the '3' across the parentheses. This distribution should result in '3y + 21'. The equation then becomes '3y + 21 – 2y = 8'. This step is a straightforward application of the distributive property, a fundamental concept in algebra. The distributive property states that a(b + c) = ab + ac, and Ernesto correctly applies this property in this step. The accuracy of this expansion is crucial, as any error here will propagate through the rest of the solution. The resulting equation, '3y + 21 – 2y = 8', is now simplified and ready for further manipulation to isolate the variable 'y'. This step is a necessary step in solving for 'y', and Ernesto's correct execution demonstrates his understanding of basic algebraic principles. The ability to accurately expand expressions is a crucial skill in algebra, and this step showcases Ernesto's proficiency in this area.

  • Step 4: y + 21 = 8

    Step 4 involves simplifying the equation '3y + 21 – 2y = 8'. By combining the '3y' and '-2y' terms, Ernesto correctly simplifies the equation to 'y + 21 = 8'. This step is a straightforward application of combining like terms, a fundamental concept in algebra. Combining like terms involves adding or subtracting terms that have the same variable raised to the same power. In this case, the '3y' and '-2y' terms are like terms, and their combination results in 'y'. The accuracy of this simplification is crucial, as any error here will lead to an incorrect solution. The resulting equation, 'y + 21 = 8', is now further simplified and closer to isolating the variable 'y'. This step is a necessary step in solving for 'y', and Ernesto's correct execution demonstrates his understanding of basic algebraic principles. The ability to accurately simplify equations by combining like terms is a crucial skill in algebra, and this step showcases Ernesto's proficiency in this area.

  • Step 5: y = -13

    In Step 5, Ernesto isolates 'y' by subtracting 21 from both sides of the equation 'y + 21 = 8'. This operation should result in 'y = 8 - 21', which simplifies to 'y = -13'. This step is a straightforward application of the subtraction property of equality, a fundamental concept in algebra. The subtraction property of equality states that if you subtract the same value from both sides of an equation, the equation remains balanced. In this case, subtracting 21 from both sides isolates 'y' on the left side of the equation. The accuracy of this subtraction is crucial, as any error here will lead to an incorrect solution. The resulting value, 'y = -13', is a potential solution for 'y', and it needs to be substituted back into the original equations to verify its correctness. This step is a necessary step in solving for 'y', and Ernesto's correct execution demonstrates his understanding of basic algebraic principles. The ability to accurately isolate variables using algebraic operations is a crucial skill in algebra, and this step showcases Ernesto's proficiency in this area.

  • Step 6: Substitute y = -13 into x = y + 7

    To find the value of 'x', we substitute 'y = -13' into the equation 'x = y + 7'. This gives us 'x = -13 + 7', which simplifies to 'x = -6'. This step is a direct application of the substitution method, where the value of one variable is substituted into an equation to solve for the other variable. The accuracy of this substitution and simplification is crucial, as any error here will lead to an incorrect value for 'x'. The resulting value, 'x = -6', is a potential solution for 'x', and it needs to be verified along with the value of 'y' in the original equations to ensure the correctness of the solution. This step is a necessary step in solving for 'x', and its correct execution demonstrates an understanding of the substitution method. The ability to accurately substitute values and simplify expressions is a crucial skill in algebra, and this step showcases proficiency in this area.

The Error: Pinpointing the Mistake

Upon careful examination of Ernesto's steps, the error is Step 3: 3y + 7 – 2y = 8. The mistake lies in the incorrect distribution of the 3 in the expression 3(y + 7). It should be 3y + 21, not 3y + 7. This seemingly small error has a cascading effect, leading to an incorrect solution for the system of equations. The distributive property is a fundamental concept in algebra, and a mistake in its application can significantly alter the outcome of a problem. This error highlights the importance of meticulousness and careful attention to detail when performing algebraic manipulations. The correct distribution would have resulted in the equation 3y + 21 – 2y = 8, which would have led to the correct solution for the system. This error serves as a valuable lesson in the importance of double-checking each step in a mathematical solution to ensure accuracy.

The Correct Solution

To solve the system correctly, we follow these steps:

  1. Isolate x in the first equation: x – y = 7 => x = y + 7

  2. Substitute the expression for x into the second equation: 3(y + 7) – 2y = 8

  3. Distribute and simplify: 3y + 21 – 2y = 8

  4. Combine like terms: y + 21 = 8

  5. Solve for y: y = 8 – 21 => y = -13

  6. Substitute y = -13 into x = y + 7: x = -13 + 7 => x = -6

Thus, the correct solution is x = -6 and y = -13. This solution can be verified by substituting these values back into the original equations. Substituting x = -6 and y = -13 into the first equation, x – y = 7, we get -6 – (-13) = 7, which simplifies to -6 + 13 = 7, which is true. Substituting x = -6 and y = -13 into the second equation, 3x – 2y = 8, we get 3(-6) – 2(-13) = 8, which simplifies to -18 + 26 = 8, which is also true. Therefore, the solution x = -6 and y = -13 satisfies both equations and is the correct solution to the system. This verification step is crucial to ensure the accuracy of the solution and to catch any potential errors made during the solving process. The ability to verify solutions is a valuable skill in mathematics, as it provides confidence in the correctness of the answer.

Conclusion: The Importance of Precision

Ernesto's attempt to solve the system of equations highlights the critical importance of precision in mathematical problem-solving. A single error, in this case, an incorrect distribution, can lead to a completely wrong answer. The correct application of the substitution method, with careful attention to algebraic details, is essential for finding the accurate solution. This example underscores the need for students to not only understand the underlying concepts but also to practice meticulous execution to avoid common pitfalls. The ability to identify and correct errors is a crucial skill in mathematics, and this analysis provides a practical example of how to develop that skill. By carefully reviewing each step and understanding the potential sources of error, students can improve their problem-solving abilities and achieve greater success in mathematics. The lesson learned from Ernesto's mistake is that even seemingly small errors can have significant consequences, and that careful attention to detail is paramount in mathematical problem-solving. This principle applies not only to solving systems of equations but also to all areas of mathematics and beyond.