Lily Begins Solving The Equation 4 ( X − 1 ) − X = 3 ( X + 5 ) − 11 4(x-1)-x=3(x+5)-11 4 ( X − 1 ) − X = 3 ( X + 5 ) − 11 . Her Work Is Shown Below.${ \begin{array}{c} 4(x-1)-x=3(x+5)-11 \ 4x-4-x=3x+15-11 \ 3x-4=3x+4 \end{array} }$How Can Her Partial Solution Be Interpreted?A. The Equation Has One

by ADMIN 300 views

Understanding the Basics of Linear Equations

Linear equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable.

Interpreting Lily's Partial Solution

Lily's partial solution to the equation 4(x−1)−x=3(x+5)−114(x-1)-x=3(x+5)-11 is shown below:

{ \begin{array}{c} 4(x-1)-x=3(x+5)-11 \\ 4x-4-x=3x+15-11 \\ 3x-4=3x+4 \end{array} \}

At first glance, it may seem like Lily's solution is correct, but upon closer inspection, we can see that there is a mistake. The equation 3x−4=3x+43x-4=3x+4 is not a valid solution, as it implies that −4=4-4=4, which is a contradiction.

What Went Wrong?

So, what went wrong in Lily's solution? The problem lies in the fact that she did not follow the correct order of operations. When simplifying the equation, she should have combined like terms and then isolated the variable. Instead, she seems to have randomly added and subtracted terms, which led to the incorrect solution.

A Step-by-Step Solution

Let's take a closer look at the original equation and solve it step by step:

4(x−1)−x=3(x+5)−114(x-1)-x=3(x+5)-11

First, we need to distribute the coefficients to the terms inside the parentheses:

4x−4−x=3x+15−114x-4-x=3x+15-11

Next, we can combine like terms:

3x−4=3x+43x-4=3x+4

Now, we can see that the equation is not solvable, as it implies that −4=4-4=4, which is a contradiction. This means that the original equation has no solution.

Conclusion

In conclusion, Lily's partial solution to the equation 4(x−1)−x=3(x+5)−114(x-1)-x=3(x+5)-11 is incorrect. She did not follow the correct order of operations, and her solution implies a contradiction. A step-by-step solution to the equation reveals that it has no solution. This highlights the importance of following the correct procedures when solving linear equations.

Common Mistakes to Avoid

When solving linear equations, it's essential to avoid common mistakes, such as:

  • Not following the correct order of operations
  • Not combining like terms
  • Not isolating the variable
  • Not checking for contradictions

By following these tips, you can ensure that your solutions are accurate and correct.

Real-World Applications

Linear equations have numerous real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects under constant acceleration.
  • Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Linear equations are used to model economic systems and make predictions about future trends.

Final Thoughts

In conclusion, solving linear equations requires attention to detail and a thorough understanding of the correct procedures. By following the steps outlined in this article, you can ensure that your solutions are accurate and correct. Remember to avoid common mistakes, such as not following the correct order of operations, and always check for contradictions. With practice and patience, you can become proficient in solving linear equations and apply them to real-world problems.

Additional Resources

For further practice and review, you can try the following resources:

  • Khan Academy: Linear Equations
  • Mathway: Linear Equations
  • Wolfram Alpha: Linear Equations

Discussion Questions

  1. What are some common mistakes to avoid when solving linear equations?
  2. How can you ensure that your solutions are accurate and correct?
  3. What are some real-world applications of linear equations?

Answer Key

  1. Not following the correct order of operations, not combining like terms, not isolating the variable, and not checking for contradictions.
  2. By following the steps outlined in this article and practicing regularly.
  3. Physics, engineering, economics, and many other fields.
    Frequently Asked Questions: Solving Linear Equations =====================================================

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable.

Q: What are some common mistakes to avoid when solving linear equations?

A: Some common mistakes to avoid when solving linear equations include:

  • Not following the correct order of operations
  • Not combining like terms
  • Not isolating the variable
  • Not checking for contradictions

Q: How can I ensure that my solutions are accurate and correct?

A: To ensure that your solutions are accurate and correct, follow these steps:

  1. Read the problem carefully and understand what is being asked.
  2. Write down the equation and identify the variable(s) and constants.
  3. Follow the correct order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).
  4. Combine like terms and simplify the equation.
  5. Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
  6. Check for contradictions and ensure that the solution is valid.

Q: What are some real-world applications of linear equations?

A: Linear equations have numerous real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects under constant acceleration.
  • Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Linear equations are used to model economic systems and make predictions about future trends.

Q: How can I practice solving linear equations?

A: There are many resources available to help you practice solving linear equations, including:

  • Khan Academy: Linear Equations
  • Mathway: Linear Equations
  • Wolfram Alpha: Linear Equations
  • Online practice problems and worksheets
  • Textbooks and study guides

Q: What are some tips for solving linear equations?

A: Some tips for solving linear equations include:

  • Read the problem carefully and understand what is being asked.
  • Use the correct order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).
  • Combine like terms and simplify the equation.
  • Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
  • Check for contradictions and ensure that the solution is valid.

Q: How can I check my work when solving linear equations?

A: To check your work when solving linear equations, follow these steps:

  1. Read the problem carefully and understand what is being asked.
  2. Write down the equation and identify the variable(s) and constants.
  3. Follow the correct order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).
  4. Combine like terms and simplify the equation.
  5. Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
  6. Check for contradictions and ensure that the solution is valid.

Q: What are some common mistakes to look out for when solving linear equations?

A: Some common mistakes to look out for when solving linear equations include:

  • Not following the correct order of operations
  • Not combining like terms
  • Not isolating the variable
  • Not checking for contradictions

Q: How can I use linear equations in real-world applications?

A: Linear equations can be used in a variety of real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects under constant acceleration.
  • Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Linear equations are used to model economic systems and make predictions about future trends.

Q: What are some resources for learning more about linear equations?

A: Some resources for learning more about linear equations include:

  • Khan Academy: Linear Equations
  • Mathway: Linear Equations
  • Wolfram Alpha: Linear Equations
  • Online practice problems and worksheets
  • Textbooks and study guides

Q: How can I get help if I'm struggling with linear equations?

A: If you're struggling with linear equations, there are many resources available to help you, including:

  • Online tutoring services
  • Math tutors
  • Study groups
  • Online forums and communities
  • Textbooks and study guides