Solve For F F F : 6 F + 9 G = 3 G + F 6f + 9g = 3g + F 6 F + 9 G = 3 G + F A. F = − 8 G 3 F = \frac{-8g}{3} F = 3 − 8 G ​ B. F = − 6 G 5 F = \frac{-6g}{5} F = 5 − 6 G ​ C. F = − 5 G 6 F = \frac{-5g}{6} F = 6 − 5 G ​ D. F = 12 G 7 F = \frac{12g}{7} F = 7 12 G ​

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Introduction

In algebra, solving for a variable means isolating that variable on one side of the equation. In this article, we will focus on solving for the variable $f$ in the given equation $6f + 9g = 3g + f$. We will use algebraic techniques to isolate $f$ and find its value in terms of $g$.

The Given Equation

The given equation is:

$$6f + 9g = 3g + f$$

Our goal is to solve for $f$, which means we need to isolate $f$ on one side of the equation.

Step 1: Subtract $f$ from Both Sides

To start solving for $f$, we need to get all the terms involving $f$ on one side of the equation. We can do this by subtracting $f$ from both sides of the equation:

$$6f - f + 9g = 3g + f - f$$

This simplifies to:

$$5f + 9g = 3g$$

Step 2: Subtract $9g$ from Both Sides

Next, we need to get rid of the term $9g$ on the left side of the equation. We can do this by subtracting $9g$ from both sides of the equation:

$$5f + 9g - 9g = 3g - 9g$$

This simplifies to:

$$5f = -6g$$

Step 3: Divide Both Sides by 5

Finally, we need to isolate $f$ by dividing both sides of the equation by 5:

$$\frac{5f}{5} = \frac{-6g}{5}$$

This simplifies to:

$$f = \frac{-6g}{5}$$

Conclusion

In this article, we solved for the variable $f$ in the given equation $6f + 9g = 3g + f$. We used algebraic techniques to isolate $f$ and found its value in terms of $g$. The final answer is:

$$f = \frac{-6g}{5}$$

This is option B in the given multiple-choice question.

Comparison with Other Options

Let's compare our answer with the other options:

• Option A: $f = \frac{-8g}{3}$
• Option C: $f = \frac{-5g}{6}$
• Option D: $f = \frac{12g}{7}$

Our answer, $f = \frac{-6g}{5}$, is different from all the other options. This means that option B is the correct answer.

Real-World Applications

Solving for a variable like $f$ has many real-world applications. For example, in physics, we often need to solve for variables like position, velocity, and acceleration. In economics, we need to solve for variables like supply and demand. In engineering, we need to solve for variables like stress and strain.

Tips and Tricks

Here are some tips and tricks to help you solve for a variable like $f$:

• Always start by isolating the variable you want to solve for.
• Use algebraic techniques like addition, subtraction, multiplication, and division to get rid of terms involving the variable.
• Be careful when dividing both sides of the equation by a term that may be equal to zero.
• Check your answer by plugging it back into the original equation.

Conclusion

In this article, we solved for the variable $f$ in the given equation $6f + 9g = 3g + f$. We used algebraic techniques to isolate $f$ and found its value in terms of $g$. The final answer is:

$$f = \frac{-6g}{5}$$

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Introduction

In our previous article, we solved for the variable $f$ in the given equation $6f + 9g = 3g + f$. We used algebraic techniques to isolate $f$ and found its value in terms of $g$. In this article, we will provide a Q&A guide to help you understand the concept of solving for a variable like $f$.

Q: What is solving for a variable?

A: Solving for a variable means isolating that variable on one side of the equation. In other words, it means finding the value of the variable in terms of other variables or constants.

Q: Why is solving for a variable important?

A: Solving for a variable is important because it allows us to find the value of the variable in terms of other variables or constants. This is useful in many real-world applications, such as physics, economics, and engineering.

Q: How do I solve for a variable like $f$?

A: To solve for a variable like $f$, you need to follow these steps:

1. Isolate the variable you want to solve for.
2. Use algebraic techniques like addition, subtraction, multiplication, and division to get rid of terms involving the variable.
3. Be careful when dividing both sides of the equation by a term that may be equal to zero.
4. Check your answer by plugging it back into the original equation.

Q: What are some common algebraic techniques used to solve for a variable?

A: Some common algebraic techniques used to solve for a variable include:

• Addition and subtraction: These techniques are used to get rid of terms involving the variable.
• Multiplication and division: These techniques are used to isolate the variable.
• Distributive property: This technique is used to expand expressions and simplify equations.
• Inverse operations: This technique is used to undo operations and isolate the variable.

Q: How do I check my answer?

A: To check your answer, you need to plug it back into the original equation and see if it is true. If it is true, then your answer is correct. If it is not true, then you need to go back and recheck your work.

Q: What are some common mistakes to avoid when solving for a variable?

A: Some common mistakes to avoid when solving for a variable include:

• Not isolating the variable correctly.
• Not using the correct algebraic techniques.
• Not checking your answer.
• Not being careful when dividing both sides of the equation by a term that may be equal to zero.

Q: How do I apply solving for a variable in real-world applications?

A: Solving for a variable is used in many real-world applications, such as:

• Physics: Solving for variables like position, velocity, and acceleration is used to describe the motion of objects.
• Economics: Solving for variables like supply and demand is used to understand the behavior of markets.
• Engineering: Solving for variables like stress and strain is used to design and build structures.

Conclusion

In this article, we provided a Q&A guide to help you understand the concept of solving for a variable like $f$. We discussed the importance of solving for a variable, common algebraic techniques used to solve for a variable, and how to check your answer. We also discussed some common mistakes to avoid when solving for a variable and how to apply solving for a variable in real-world applications.