What Is The Solution To The Equation $\frac{x}{3}+\frac{x}{6}=\frac{7}{2}$?A. $x=\frac{3}{2}$ B. $x=\frac{7}{3}$ C. $x=3$ D. $x=7$

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Introduction

Solving equations is a fundamental concept in mathematics, and it is essential to understand how to approach and solve various types of equations. In this article, we will focus on solving a specific equation, $\frac{x}{3}+\frac{x}{6}=\frac{7}{2}$, and explore the different methods and techniques used to find the solution.

Understanding the Equation

The given equation is a linear equation, which means it can be solved using basic algebraic operations. The equation is $\frac{x}{3}+\frac{x}{6}=\frac{7}{2}$. To solve this equation, we need to isolate the variable $x$.

Step 1: Multiply Both Sides by the Least Common Multiple (LCM)

The first step in solving this equation is to eliminate the fractions by multiplying both sides by the least common multiple (LCM) of the denominators. In this case, the LCM of 3 and 6 is 6. Multiplying both sides by 6 gives us:

$$6\left(\frac{x}{3}+\frac{x}{6}\right)=6\left(\frac{7}{2}\right)$$

Step 2: Distribute the Multiplication

Now, we need to distribute the multiplication to simplify the equation. This gives us:

$$2x+x=21$$

Step 3: Combine Like Terms

Next, we combine like terms to simplify the equation further. This gives us:

$$3x=21$$

Step 4: Divide Both Sides by 3

Finally, we divide both sides by 3 to isolate the variable $x$. This gives us:

$$x=\frac{21}{3}$$

Step 5: Simplify the Fraction

We can simplify the fraction $\frac{21}{3}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This gives us:

$$x=7$$

Conclusion

In conclusion, the solution to the equation $\frac{x}{3}+\frac{x}{6}=\frac{7}{2}$ is $x=7$. This is the correct answer among the options provided.

Answer Key

The correct answer is:

• D. $x=7$

Discussion

This equation is a simple linear equation, and it can be solved using basic algebraic operations. The key steps in solving this equation are multiplying both sides by the LCM, distributing the multiplication, combining like terms, and finally dividing both sides by 3 to isolate the variable $x$. This equation is a great example of how to approach and solve linear equations.

Related Topics

• Solving linear equations
• Algebraic operations
• Least common multiple (LCM)
• Distributing multiplication
• Combining like terms
• Dividing both sides by a constant

Final Thoughts

Solving equations is a fundamental concept in mathematics, and it is essential to understand how to approach and solve various types of equations. In this article, we focused on solving a specific equation, $\frac{x}{3}+\frac{x}{6}=\frac{7}{2}$, and explored the different methods and techniques used to find the solution. By following the steps outlined in this article, you can solve similar equations and become more confident in your ability to approach and solve linear equations.

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Introduction

In our previous article, we solved the equation $\frac{x}{3}+\frac{x}{6}=\frac{7}{2}$ and found that the solution is $x=7$. However, we understand that some readers may still have questions or doubts about the solution. In this article, we will address some of the most frequently asked questions (FAQs) about solving this equation.

Q: What is the least common multiple (LCM) of 3 and 6?

A: The least common multiple (LCM) of 3 and 6 is 6. This is because 6 is the smallest number that both 3 and 6 can divide into evenly.

Q: Why do we multiply both sides of the equation by 6?

A: We multiply both sides of the equation by 6 to eliminate the fractions. By doing so, we can simplify the equation and make it easier to solve.

Q: Can we use a different method to solve this equation?

A: Yes, there are other methods to solve this equation. However, the method we used in our previous article is one of the most straightforward and efficient ways to solve it.

Q: What if the equation had a different denominator, such as 4 or 8?

A: If the equation had a different denominator, we would need to find the least common multiple (LCM) of the new denominator and the original denominator. We would then multiply both sides of the equation by the LCM to eliminate the fractions.

Q: Can we use a calculator to solve this equation?

A: Yes, we can use a calculator to solve this equation. However, it's always a good idea to understand the steps involved in solving the equation and to check our work to ensure that we have the correct solution.

Q: What if we made a mistake in solving the equation?

A: If we made a mistake in solving the equation, we can go back and recheck our work. We can also try solving the equation again using a different method to see if we get the same solution.

Q: Is there a shortcut to solving this equation?

A: While there may not be a shortcut to solving this equation, we can use some algebraic properties to simplify the equation and make it easier to solve.

Q: Can we use this method to solve other types of equations?

A: Yes, we can use this method to solve other types of equations that involve fractions. However, we may need to adjust the method slightly depending on the specific equation.

Conclusion

In conclusion, solving the equation $\frac{x}{3}+\frac{x}{6}=\frac{7}{2}$ requires a clear understanding of algebraic operations and the concept of least common multiple (LCM). By following the steps outlined in this article, you can solve similar equations and become more confident in your ability to approach and solve linear equations.

Related Topics

• Solving linear equations
• Algebraic operations
• Least common multiple (LCM)
• Distributing multiplication
• Combining like terms
• Dividing both sides by a constant

Final Thoughts

Solving equations is a fundamental concept in mathematics, and it is essential to understand how to approach and solve various types of equations. By practicing and reviewing different types of equations, you can become more confident in your ability to solve them and apply mathematical concepts to real-world problems.