Common Mistakes in Algebra

Oct 13, 2025 by talynt.org 58 views

Understanding the Problem

Janelle attempted to expand the given expression, but unfortunately, she made some errors. To identify the mistakes, we need to carefully analyze the expression and the correct expansion.

The Given Expression

The expression given by Janelle is:

-4\left(-3x+\frac{2}{7}\right)=-12x-3\frac{5}{7}

Correct Expansion

To expand the given expression, we need to apply the distributive property, which states that for any real numbers a, b, and c:

a(b+c)=ab+ac

Using this property, we can expand the given expression as follows:

-4\left(-3x+\frac{2}{7}\right)=-4(-3x)+(-4)\left(\frac{2}{7}\right)

Now, we can simplify each term:

-4(-3x)=12x

(-4)\left(\frac{2}{7}\right)=-\frac{8}{7}

Combining the two terms, we get:

12x-\frac{8}{7}

Comparing with Janelle's Answer

Now, let's compare Janelle's answer with the correct expansion:

-12x-3\frac{5}{7}

We can see that Janelle's answer is incorrect. The correct expansion is:

12x-\frac{8}{7}

Identifying the Errors

Based on the correct expansion, we can identify the errors made by Janelle:

The first term should be positive, not negative. The correct expansion has a positive 12x term.
The second term should be negative, not positive. The correct expansion has a negative -8/7 term.
The correct expansion does not have a 3/5 term. The correct expansion only has a -8/7 term.

Conclusion

In conclusion, Janelle made three errors in expanding the given expression:

The first term should be positive.
The second term should be negative.
The correct expansion does not have a 3/5 term.

By carefully analyzing the expression and the correct expansion, we can identify the mistakes made by Janelle and provide a clear understanding of the correct solution.

Key Takeaways

The distributive property is a fundamental concept in algebra that allows us to expand expressions.
When expanding expressions, we need to carefully apply the distributive property to ensure accuracy.
Identifying errors in algebraic expressions requires a thorough understanding of the concepts and a careful analysis of the given expression.

Common Mistakes in Algebra

Neglecting to apply the distributive property.
Incorrectly applying the distributive property.
Failing to simplify expressions.

By being aware of these common mistakes, we can avoid making similar errors and provide accurate solutions to algebraic expressions.

Real-World Applications

Algebraic expressions are used in various real-world applications, such as physics, engineering, and economics.
Understanding algebraic expressions is crucial in solving problems related to motion, forces, and energy.
Algebraic expressions are also used in financial modeling, where they help in predicting stock prices and returns.

By understanding algebraic expressions and their applications, we can develop a deeper appreciation for the subject and its relevance in real-world scenarios.

Final Thoughts

Understanding the Problem

In our previous article, we discussed Janelle's mistake in expanding the given expression. We identified the errors she made and provided a clear understanding of the correct solution. In this article, we will answer some frequently asked questions related to the topic.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions. It states that for any real numbers a, b, and c:

a(b+c)=ab+ac

Q: How do I apply the distributive property?

A: To apply the distributive property, you need to multiply each term inside the parentheses by the coefficient outside the parentheses. In the given expression, we have:

-4\left(-3x+\frac{2}{7}\right)

We need to multiply each term inside the parentheses by -4:

-4(-3x)=-12x

(-4)\left(\frac{2}{7}\right)=-\frac{8}{7}

Q: What are the common mistakes in algebra?

A: Some common mistakes in algebra include:

Neglecting to apply the distributive property.
Incorrectly applying the distributive property.
Failing to simplify expressions.

Q: How do I identify errors in algebraic expressions?

A: To identify errors in algebraic expressions, you need to carefully analyze the expression and compare it with the correct solution. In Janelle's case, we identified the errors by comparing her answer with the correct expansion:

-12x-3\frac{5}{7}

We found that the first term should be positive, not negative, and the second term should be negative, not positive.

Q: What are the real-world applications of algebraic expressions?

A: Algebraic expressions are used in various real-world applications, such as:

Physics: Algebraic expressions are used to describe the motion of objects, forces, and energy.
Engineering: Algebraic expressions are used to design and optimize systems, such as bridges and buildings.
Economics: Algebraic expressions are used to model economic systems and predict stock prices and returns.

Q: How can I improve my problem-solving skills in algebra?

A: To improve your problem-solving skills in algebra, you need to:

Practice regularly: Practice solving algebraic expressions and equations to develop your skills.
Understand the concepts: Make sure you understand the concepts and theorems in algebra.
Analyze the expressions: Carefully analyze the expressions and identify the errors.

Q: What are some tips for expanding algebraic expressions?

A: Some tips for expanding algebraic expressions include:

Apply the distributive property carefully.
Simplify the expressions as you go.
Check your work by comparing it with the correct solution.

Conclusion

In conclusion, Janelle's mistake in expanding the given expression highlights the importance of carefully analyzing expressions and applying the distributive property accurately. By understanding the correct expansion and identifying the errors made by Janelle, we can develop a stronger foundation in algebra and improve our problem-solving skills. We hope this Q&A article has provided you with a better understanding of the topic and has helped you to improve your skills in algebra.

Key Takeaways

The distributive property is a fundamental concept in algebra that allows us to expand expressions.
Algebraic expressions are used in various real-world applications, such as physics, engineering, and economics.
Understanding algebraic expressions and their applications is crucial in developing problem-solving skills.

Real-World Applications

Algebraic expressions are used in physics to describe the motion of objects, forces, and energy.
Algebraic expressions are used in engineering to design and optimize systems, such as bridges and buildings.
Algebraic expressions are used in economics to model economic systems and predict stock prices and returns.