A Student Provided The Following Solution For Solving An Equation. What Mistake Did They Make?Original Equation: $ 3(a-2)+14=8-6a }$Solution $[ \begin{aligned 3a-2+14 & =8-6a \ 3a+12 & =8-6a \ 9a+12 & =8 \ 9a & =-4 \ a &

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Introduction

Solving equations is a fundamental concept in mathematics, and it's essential to understand the correct procedures to arrive at the solution. However, students often make mistakes in their calculations, which can lead to incorrect answers. In this article, we will analyze a student's solution to an equation and identify the mistake they made.

The Original Equation

The original equation provided by the student is:

3(a−2)+14=8−6a{ 3(a-2)+14=8-6a }

This equation involves a linear expression inside the parentheses, which needs to be simplified and then solved for the variable 'a'.

The Student's Solution

The student's solution is as follows:

3a−2+14=8−6a3a+12=8−6a9a+12=89a=−4a{ \begin{aligned} 3a-2+14 & =8-6a \\ 3a+12 & =8-6a \\ 9a+12 & =8 \\ 9a & =-4 \\ a & \end{aligned} }

Identifying the Mistake

Upon analyzing the student's solution, we can see that they made a mistake in the second step. The correct expansion of the left-hand side of the equation should be:

3(a−2)+14=3a−6+14{ 3(a-2)+14 = 3a - 6 + 14 }

However, the student incorrectly simplified the expression to:

3a+12=8−6a{ 3a+12 = 8-6a }

This mistake led to a chain of incorrect calculations, resulting in the final answer of:

a=−49{ a = -\frac{4}{9} }

Correct Solution

To solve the original equation correctly, we need to follow the correct procedures:

  1. Expand the left-hand side of the equation:

3(a−2)+14=3a−6+14{ 3(a-2)+14 = 3a - 6 + 14 }

  1. Simplify the expression:

3a−6+14=3a+8{ 3a - 6 + 14 = 3a + 8 }

  1. Rewrite the equation with the simplified expression:

3a+8=8−6a{ 3a + 8 = 8 - 6a }

  1. Add 6a to both sides of the equation to isolate the variable 'a':

3a+6a+8=8{ 3a + 6a + 8 = 8 }

  1. Combine like terms:

9a+8=8{ 9a + 8 = 8 }

  1. Subtract 8 from both sides of the equation:

9a=−8{ 9a = -8 }

  1. Divide both sides of the equation by 9 to solve for 'a':

a=−89{ a = -\frac{8}{9} }

Conclusion

In conclusion, the student made a mistake in the second step of their solution, which led to a chain of incorrect calculations. By following the correct procedures and simplifying the expression correctly, we arrived at the correct solution of:

a=−89{ a = -\frac{8}{9} }

This analysis highlights the importance of careful attention to detail and following the correct procedures when solving equations.

Common Mistakes in Solving Equations

There are several common mistakes that students make when solving equations, including:

  • Incorrectly expanding or simplifying expressions
  • Failing to isolate the variable correctly
  • Not following the order of operations
  • Making arithmetic errors

By being aware of these common mistakes, students can take steps to avoid them and arrive at the correct solution.

Tips for Solving Equations

Here are some tips for solving equations:

  • Read the equation carefully and understand what is being asked
  • Follow the order of operations (PEMDAS)
  • Simplify expressions correctly
  • Isolate the variable correctly
  • Check your work by plugging the solution back into the original equation

By following these tips, students can improve their skills in solving equations and arrive at the correct solution.

Final Thoughts

Solving equations is a critical skill in mathematics, and it's essential to understand the correct procedures to arrive at the solution. By analyzing a student's mistake and identifying the error, we can learn from their mistake and improve our own skills in solving equations. Remember to always follow the correct procedures, simplify expressions correctly, and check your work to ensure that you arrive at the correct solution.

Introduction

In our previous article, we analyzed a student's solution to an equation and identified the mistake they made. In this article, we will provide a Q&A section to help students and teachers understand the correct procedures for solving equations and to address common questions and concerns.

Q&A Section

Q: What is the most common mistake students make when solving equations?

A: The most common mistake students make when solving equations is incorrectly expanding or simplifying expressions. This can lead to a chain of incorrect calculations and a final answer that is incorrect.

Q: How can I avoid making mistakes when solving equations?

A: To avoid making mistakes when solving equations, make sure to:

  • Read the equation carefully and understand what is being asked
  • Follow the order of operations (PEMDAS)
  • Simplify expressions correctly
  • Isolate the variable correctly
  • Check your work by plugging the solution back into the original equation

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when solving an equation. The acronym PEMDAS stands for:

  • P: Parentheses (evaluate expressions inside parentheses first)
  • E: Exponents (evaluate any exponential expressions next)
  • M: Multiplication and Division (evaluate any multiplication and division operations from left to right)
  • A: Addition and Subtraction (finally, evaluate any addition and subtraction operations from left to right)

Q: How do I simplify expressions correctly?

A: To simplify expressions correctly, make sure to:

  • Combine like terms (e.g., 2x + 3x = 5x)
  • Eliminate any unnecessary parentheses or brackets
  • Use the distributive property (e.g., a(b + c) = ab + ac)

Q: What is the distributive property?

A: The distributive property is a rule that allows us to expand expressions by multiplying each term inside the parentheses by the term outside the parentheses. For example:

a(b + c) = ab + ac

Q: How do I isolate the variable correctly?

A: To isolate the variable correctly, make sure to:

  • Add or subtract the same value to both sides of the equation to eliminate any constants
  • Multiply or divide both sides of the equation by the same value to eliminate any coefficients

Q: What is the final answer to the original equation?

A: The final answer to the original equation is:

a = -\frac{8}{9}

Q: Can I plug the solution back into the original equation to check my work?

A: Yes, you can plug the solution back into the original equation to check your work. This is a great way to verify that your solution is correct.

Conclusion

In conclusion, solving equations is a critical skill in mathematics, and it's essential to understand the correct procedures to arrive at the solution. By following the correct procedures, simplifying expressions correctly, and checking your work, you can avoid making mistakes and arrive at the correct solution.

Common Questions and Concerns

Here are some common questions and concerns that students and teachers may have when solving equations:

  • Q: What if I get stuck on a problem and don't know how to solve it? A: Don't worry! If you get stuck on a problem, try breaking it down into smaller steps or asking for help from a teacher or classmate.
  • Q: What if I make a mistake and don't know how to correct it? A: Don't worry! If you make a mistake, try to identify the error and correct it. If you're still unsure, ask for help from a teacher or classmate.
  • Q: What if I'm not sure if my solution is correct? A: Don't worry! If you're not sure if your solution is correct, try plugging it back into the original equation to check your work.

Final Thoughts

Solving equations is a critical skill in mathematics, and it's essential to understand the correct procedures to arrive at the solution. By following the correct procedures, simplifying expressions correctly, and checking your work, you can avoid making mistakes and arrive at the correct solution. Remember to always ask for help if you're unsure, and don't be afraid to make mistakes – they're an opportunity to learn and grow!