Solving System Of Equations Using Substitution Method Error Analysis

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Ernesto embarked on a journey to solve a system of equations using the substitution method, a powerful technique in algebra. His work, as presented, offers a fascinating case study in the application of this method. Let's delve into his steps, scrutinize his calculations, and pinpoint any missteps along the way. This detailed analysis will not only help Ernesto (and our readers) understand the correct solution but also reinforce the fundamental principles of solving systems of equations.

The System of Equations

At the heart of our exploration lies the following system of equations:

x - y = 7
3x - 2y = 8

This system presents us with two linear equations, each containing two variables, x and y. Our goal is to find the values of x and y that satisfy both equations simultaneously. The substitution method, as Ernesto attempted to use, is a classic approach to tackling such problems. It involves solving one equation for one variable and then substituting that expression into the other equation, thereby reducing the system to a single equation with a single variable.

Emesto's Steps: A Critical Examination

Let's meticulously examine Ernesto's steps, highlighting both the correct moves and any potential errors:

Step 1: Isolating x

Ernesto's first step was to isolate x in the first equation. He correctly added y to both sides of the equation x - y = 7, resulting in:

x = y + 7

This step is correct and lays the foundation for the substitution method. By expressing x in terms of y, Ernesto has created an expression that can be substituted into the second equation.

Step 2: Substitution

The next logical step is to substitute the expression y + 7 for x in the second equation. Ernesto correctly performed this substitution, replacing x in the equation 3x - 2y = 8 with (y + 7):

3(y + 7) - 2y = 8

This step is also correct. The substitution has successfully transformed the second equation into an equation with only one variable, y.

Step 3: Distribute

This is the crucial stage where errors often creep in. It's essential to meticulously distribute the 3 across the parentheses. Multiplying out should give us 3 times y plus 3 times 7. Let’s see how Emesto handled it. This simplification is vital for solving for y. The accurate distribution and combination of like terms are the linchpins in reaching the correct answer. Any slip-up here will throw off the rest of the solution.

Identifying the Error and Correcting the Solution

Upon a close inspection of Emesto's attempt, the critical error likely lies in the distribution and simplification steps following the substitution. To clarify, let’s take it from Emesto’s correct step 2:

Step 2:

3(y + 7) - 2y = 8

The next step would accurately distribute the 3 across the (y + 7):

3y + 21 - 2y = 8

After that, the y terms must be correctly combined. This implies subtracting 2y from 3y:

y + 21 = 8

Next, we subtract 21 from both sides of the equation to isolate y:

y = 8 - 21
y = -13

So, the correct value for y is -13.

Now that we have the correct value for y, we can substitute it back into the equation x = y + 7 (from Step 1) to find the value of x:

x = -13 + 7
x = -6

Thus, the correct value for x is -6.

The Correct Solution

The solution to the system of equations is therefore:

x = -6
y = -13

This means that the point (-6, -13) is the intersection of the two lines represented by the equations. This pair of values for x and y will satisfy both equations in the system.

Verifying the Solution

It's always a good practice to verify the solution by substituting the values of x and y back into the original equations. Let's do that:

Equation 1:

x - y = 7
-6 - (-13) = 7
-6 + 13 = 7
7 = 7 (Correct)

Equation 2:

3x - 2y = 8
3(-6) - 2(-13) = 8
-18 + 26 = 8
8 = 8 (Correct)

Since the values x = -6 and y = -13 satisfy both equations, we have confirmed that our solution is correct.

Common Mistakes in Substitution

Ernesto's attempt, while containing an error, highlights several common pitfalls students encounter when using the substitution method. Awareness of these potential missteps can significantly improve accuracy:

  1. Incorrect Distribution: As seen in Emesto's case, errors in distributing a constant across parentheses are frequent. It's vital to ensure that every term inside the parentheses is multiplied by the constant.

  2. Sign Errors: Dealing with negative signs can be tricky. A misplaced negative sign can lead to an incorrect solution. Pay close attention to the signs of the terms when substituting and simplifying.

  3. Incorrect Substitution: Ensure that you are substituting the expression for the correct variable. Substituting for the wrong variable will not lead to a simplified equation.

  4. Arithmetic Errors: Simple arithmetic mistakes, such as addition or subtraction errors, can derail the entire process. Double-check your calculations at each step.

  5. Forgetting to Solve for Both Variables: Once you've found the value of one variable, remember to substitute it back into one of the original equations (or the equation you derived in Step 1) to solve for the other variable. A complete solution requires values for both x and y.

Best Practices for Using Substitution

To effectively use the substitution method and minimize errors, consider these best practices:

  • Choose the Easiest Variable to Isolate: Look for an equation where one variable has a coefficient of 1 or -1. This will simplify the process of isolating that variable.

  • Write Neatly and Organize Your Work: A clear and organized presentation of your steps will make it easier to spot errors. Use plenty of space and write legibly.

  • Double-Check Each Step: Before moving on, review your work to ensure that you haven't made any mistakes in substitution, distribution, or simplification.

  • Verify Your Solution: Always substitute your solution back into the original equations to check for accuracy.

  • Practice Regularly: Like any mathematical skill, proficiency in the substitution method comes with practice. Work through a variety of examples to solidify your understanding.

Conclusion

Solving systems of equations is a fundamental skill in algebra, with applications in various fields. The substitution method is a powerful tool, but it requires careful execution. By meticulously analyzing Emesto's attempt, we've identified a common error in distribution and have reinforced the importance of careful calculation and verification. By understanding the potential pitfalls and adhering to best practices, students can confidently and accurately solve systems of equations using the substitution method. Remember, mathematics is not just about finding the right answer; it's about understanding the process and developing problem-solving skills that will serve you well in various aspects of life.

Therefore, careful attention to detail, a systematic approach, and consistent verification are key to mastering this essential algebraic technique. The ability to solve systems of equations opens doors to more advanced mathematical concepts and real-world applications, making it a valuable skill to cultivate.

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