Which Number Can Each Term Of The Equation Be Multiplied By To Eliminate The Fractions Before Solving?${ 6 - \frac{3}{4}x + \frac{1}{3} = \frac{1}{2}x + 5 }$A. 2 B. 3 C. 6 D. 12

When dealing with equations that contain fractions, it can be challenging to solve them. One way to simplify the process is to eliminate the fractions before solving the equation. In this article, we will explore how to do this by multiplying each term of the equation by a specific number.

Understanding the Problem

The given equation is:

${ 6 - \frac{3}{4}x + \frac{1}{3} = \frac{1}{2}x + 5 }$

Our goal is to eliminate the fractions in this equation, making it easier to solve.

Identifying the Least Common Multiple (LCM)

To eliminate the fractions, we need to find the least common multiple (LCM) of the denominators. In this case, the denominators are 4, 3, and 2. The LCM of these numbers is 12.

Multiplying Each Term by the LCM

To eliminate the fractions, we can multiply each term of the equation by the LCM, which is 12.

${ 12 \times (6 - \frac{3}{4}x + \frac{1}{3}) = 12 \times (\frac{1}{2}x + 5) }$

This simplifies to:

${ 72 - 9x + 4 = 6x + 60 }$

Simplifying the Equation

Now that we have eliminated the fractions, we can simplify the equation by combining like terms.

${ 76 - 9x = 6x + 60 }$

Isolating the Variable

To solve for x, we need to isolate the variable on one side of the equation. We can do this by subtracting 6x from both sides and adding 9x to both sides.

${ 76 = 15x + 60 }$

Subtracting 60 from both sides gives us:

${ 16 = 15x }$

Solving for x

Finally, we can solve for x by dividing both sides by 15.

${ x = \frac{16}{15} }$

Conclusion

In this article, we learned how to eliminate fractions in an equation by multiplying each term by the least common multiple (LCM) of the denominators. We applied this technique to the given equation and simplified it to solve for x. The correct answer is:

The correct answer is D. 12

Why is this the correct answer?

The correct answer is D. 12 because it is the least common multiple (LCM) of the denominators in the given equation. Multiplying each term by 12 eliminates the fractions and allows us to simplify the equation and solve for x.

Tips and Tricks

• When dealing with equations that contain fractions, it's essential to find the least common multiple (LCM) of the denominators.
• Multiplying each term by the LCM eliminates the fractions and simplifies the equation.
• Be careful when simplifying the equation and combining like terms.
• Isolate the variable on one side of the equation to solve for x.

Common Mistakes

• Failing to find the least common multiple (LCM) of the denominators.
• Not multiplying each term by the LCM.
• Not simplifying the equation correctly.
• Not isolating the variable on one side of the equation.

Real-World Applications

Eliminating fractions in equations is a crucial skill in various fields, including:

• Algebra
• Calculus
• Physics
• Engineering

By mastering this technique, you can solve complex equations and make informed decisions in your field of study or profession.

Final Thoughts

In our previous article, we explored how to eliminate fractions in equations by multiplying each term by the least common multiple (LCM) of the denominators. However, we know that there are many more questions and concerns that you may have. In this article, we will address some of the most frequently asked questions (FAQs) about eliminating fractions in equations.

Q: What is the least common multiple (LCM) of the denominators?

A: The least common multiple (LCM) of the denominators is the smallest number that all the denominators can divide into evenly. For example, if the denominators are 4, 3, and 2, the LCM is 12.

Q: How do I find the LCM of the denominators?

A: To find the LCM of the denominators, you can use the following steps:

1. List the denominators.
2. Find the prime factors of each denominator.
3. Multiply the highest power of each prime factor together.
4. The result is the LCM of the denominators.

Q: What if the denominators have different prime factors?

A: If the denominators have different prime factors, you can still find the LCM by multiplying the highest power of each prime factor together. For example, if the denominators are 4 (2^2) and 3, the LCM is 12 (2^2 x 3).

Q: Can I use a calculator to find the LCM of the denominators?

A: Yes, you can use a calculator to find the LCM of the denominators. Most calculators have a built-in function to find the LCM.

Q: What if I have a fraction with a variable in the denominator?

A: If you have a fraction with a variable in the denominator, you can still find the LCM by treating the variable as a constant. For example, if the fraction is 1/x, the LCM is x.

Q: Can I eliminate fractions in equations with multiple variables?

A: Yes, you can eliminate fractions in equations with multiple variables by finding the LCM of the denominators and multiplying each term by it.

Q: What if I have an equation with a fraction and a variable in the numerator?

A: If you have an equation with a fraction and a variable in the numerator, you can still eliminate the fraction by finding the LCM of the denominators and multiplying each term by it.

Q: Can I use other methods to eliminate fractions in equations?

A: Yes, there are other methods to eliminate fractions in equations, such as multiplying both sides of the equation by the LCM of the denominators or using algebraic manipulations. However, multiplying each term by the LCM is often the most straightforward method.

Q: What are some common mistakes to avoid when eliminating fractions in equations?

A: Some common mistakes to avoid when eliminating fractions in equations include:

• Failing to find the LCM of the denominators.
• Not multiplying each term by the LCM.
• Not simplifying the equation correctly.
• Not isolating the variable on one side of the equation.

Q: How can I practice eliminating fractions in equations?

A: You can practice eliminating fractions in equations by working through examples and exercises. You can also use online resources, such as worksheets and practice tests, to help you improve your skills.

Conclusion

Eliminating fractions in equations is a crucial skill in mathematics and other fields. By understanding the concept of the least common multiple (LCM) and how to find it, you can simplify equations and solve for variables. We hope that this article has helped to answer some of your questions and concerns about eliminating fractions in equations. Remember to practice regularly to improve your skills and become proficient in solving equations with fractions.