Not solving for x correctly

Oct 13, 2025 by miodeya.com 85 views

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will guide you through the process of solving linear equations, using the example of Liza, who is struggling to solve a particular equation. We will examine her work, identify the mistakes, and provide a step-by-step solution to the equation.

Liza's Equation

Liza is trying to solve the following linear equation:

${ 4(x-5) + 2x = 9x - 2(4 + 2x) }$

She has completed two steps, but now she's stuck. Let's examine her work:

Liza's First Step

Liza's first step is to simplify the left-hand side of the equation:

${ 2x - 20 + 2x = 2x - 8 - 2x }$

However, there are a few issues with this step. Firstly, the equation on the left-hand side is not simplified correctly. The correct simplification would be:

${ 4(x-5) + 2x = 4x - 20 + 2x }$

Secondly, the equation on the right-hand side is not equivalent to the left-hand side. The correct right-hand side would be:

${ 2x - 8 - 2x = -8 }$

Liza's Second Step

Liza's second step is to simplify the right-hand side of the equation:

${ 6x - 20 = 6x - 8 }$

However, there are a few issues with this step. Firstly, the equation on the left-hand side is not simplified correctly. The correct simplification would be:

${ 4x - 20 + 2x = 6x - 20 }$

Secondly, the equation on the right-hand side is not equivalent to the left-hand side. The correct right-hand side would be:

${ 9x - 2(4 + 2x) = 9x - 8 - 4x }$

Correcting Liza's Mistakes

Now that we have identified Liza's mistakes, let's correct them and solve the equation.

Step 1: Simplify the Left-Hand Side

The first step is to simplify the left-hand side of the equation:

${ 4(x-5) + 2x = 4x - 20 + 2x }$

This simplifies to:

${ 6x - 20 = 6x - 20 }$

Step 2: Simplify the Right-Hand Side

The next step is to simplify the right-hand side of the equation:

${ 9x - 2(4 + 2x) = 9x - 8 - 4x }$

This simplifies to:

${ 5x - 8 = 5x - 8 }$

Step 3: Equate the Left-Hand Side and Right-Hand Side

Now that we have simplified both sides of the equation, we can equate them:

${ 6x - 20 = 5x - 8 }$

Step 4: Solve for x

The final step is to solve for x. To do this, we can add 20 to both sides of the equation:

${ 6x = 5x + 12 }$

Subtracting 5x from both sides gives us:

${ x = 12 }$

Therefore, the solution to the equation is x = 12.

Conclusion

Solving linear equations requires careful attention to detail and a step-by-step approach. By examining Liza's work and identifying her mistakes, we were able to correct her errors and solve the equation. We hope that this example has helped you to understand the process of solving linear equations and has provided you with a useful guide to follow.

Common Mistakes to Avoid

When solving linear equations, there are several common mistakes to avoid. These include:

Not simplifying the left-hand side and right-hand side correctly
Not equating the left-hand side and right-hand side correctly
Not solving for x correctly

By avoiding these common mistakes, you can ensure that you are solving linear equations correctly and accurately.

Tips and Tricks

Here are a few tips and tricks to help you solve linear equations:

Always simplify the left-hand side and right-hand side correctly
Always equate the left-hand side and right-hand side correctly
Always solve for x correctly
Use a step-by-step approach to solve the equation
Check your work carefully to avoid mistakes

By following these tips and tricks, you can become more confident and proficient in solving linear equations.

Real-World Applications

Linear equations have many real-world applications. These include:

Physics and engineering: Linear equations are used to describe the motion of objects and the behavior of physical systems.
Economics: Linear equations are used to model economic systems and make predictions about economic trends.
Computer science: Linear equations are used in computer algorithms and data analysis.
Biology: Linear equations are used to model population growth and disease spread.

By understanding how to solve linear equations, you can apply this knowledge to a wide range of real-world problems and situations.

Conclusion

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In our previous article, we provided a step-by-step guide on how to solve linear equations. In this article, we will answer some of the most frequently asked questions about solving linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (usually x) is 1. For example, 2x + 3 = 5 is a linear equation.

Q: How do I simplify a linear equation?

A: To simplify a linear equation, you need to combine like terms. Like terms are terms that have the same variable and exponent. For example, 2x + 3x can be simplified to 5x.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable (usually x). You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable (usually x) is 1. A quadratic equation is an equation in which the highest power of the variable (usually x) is 2. For example, x^2 + 3x + 2 = 0 is a quadratic equation.

Q: How do I know if an equation is linear or quadratic?

A: To determine if an equation is linear or quadratic, you need to look at the highest power of the variable (usually x). If the highest power is 1, the equation is linear. If the highest power is 2, the equation is quadratic.

Q: Can I use a calculator to solve a linear equation?

A: Yes, you can use a calculator to solve a linear equation. However, it's always a good idea to check your work by hand to make sure the calculator is giving you the correct answer.

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

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

Not simplifying the left-hand side and right-hand side correctly
Not equating the left-hand side and right-hand side correctly
Not solving for x correctly
Not checking your work carefully

Q: How do I check my work when solving a linear equation?

A: To check your work when solving a linear equation, you need to plug your answer back into the original equation and make sure it's true. For example, if you solve the equation 2x + 3 = 5 and get x = 1, you need to plug x = 1 back into the original equation and make sure it's true: 2(1) + 3 = 5.

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

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

Physics and engineering: Linear equations are used to describe the motion of objects and the behavior of physical systems.
Economics: Linear equations are used to model economic systems and make predictions about economic trends.
Computer science: Linear equations are used in computer algorithms and data analysis.
Biology: Linear equations are used to model population growth and disease spread.

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By following a step-by-step approach and avoiding common mistakes, you can ensure that you are solving linear equations correctly and accurately. We hope that this Q&A guide has helped you to understand the process of solving linear equations and has provided you with a useful resource to refer to.

Additional Resources

If you're looking for additional resources to help you solve linear equations, here are a few suggestions:

Khan Academy: Khan Academy has a comprehensive video series on solving linear equations.
Mathway: Mathway is an online math problem solver that can help you solve linear equations.
Wolfram Alpha: Wolfram Alpha is a powerful online calculator that can help you solve linear equations.

We hope that this Q&A guide has been helpful in answering your questions about solving linear equations. If you have any further questions, please don't hesitate to ask.