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

Mar 9, 2025 by ADMIN 199 views

Solving for f: A Step-by-Step Guide to Isolating the Variable

In algebra, solving for a variable means isolating that variable on one side of the equation. This is a crucial skill in mathematics, as it allows us to find the value of a variable in a given equation. In this article, we will focus on solving for the variable f in the equation 6f + 9g = 3g + f.

Understanding the Equation

The given equation is 6f + 9g = 3g + f. To solve for f, we need to isolate f on one side of the equation. This means that we need to get rid of the terms that involve g, so that we are left with only the terms that involve f.

Step 1: Subtract 3g from Both Sides

To start solving for f, we need to get rid of the term 3g on the right-hand side of the equation. We can do this by subtracting 3g from both sides of the equation.

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

This simplifies to:

6f + 6g = f

Step 2: Subtract f from Both Sides

Next, we need to get rid of the term f on the left-hand side of the equation. We can do this by subtracting f from both sides of the equation.

6f - f + 6g = f - f

This simplifies to:

5f + 6g = 0

Step 3: Subtract 6g from Both Sides

Now, we need to get rid of the term 6g on the left-hand side of the equation. We can do this by subtracting 6g from both sides of the equation.

5f + 6g - 6g = 0 - 6g

This simplifies to:

5f = -6g

Step 4: Divide Both Sides by 5

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

(5f)/5 = (-6g)/5

This simplifies to:

f = -6g/5

In conclusion, we have solved for the variable f in the equation 6f + 9g = 3g + f. By following the steps outlined above, we have isolated f on one side of the equation and found that f = -6g/5.

The correct answer is:

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

This problem is a great example of how to solve for a variable in a linear equation. By following the steps outlined above, we can isolate the variable and find its value. This skill is essential in mathematics, as it allows us to solve a wide range of problems involving linear equations.

When solving for a variable, it's essential to isolate the variable on one side of the equation.
Use inverse operations to get rid of terms that involve the variable.
Be careful when subtracting or adding terms to both sides of the equation.
Check your work by plugging the solution back into the original equation.

Solve for x in the equation 2x + 5 = 3x - 2.
Solve for y in the equation 4y - 3 = 2y + 1.
Solve for z in the equation 3z + 2 = z - 4.

In conclusion, solving for a variable is a crucial skill in mathematics. By following the steps outlined above, we can isolate the variable and find its value. This skill is essential in mathematics, as it allows us to solve a wide range of problems involving linear equations.
Solving for f: A Q&A Guide to Isolating the Variable

In our previous article, we solved for the variable f in the equation 6f + 9g = 3g + f. We followed a step-by-step approach to isolate f on one side of the equation and found that f = -6g/5. In this article, we will answer some common questions related to solving for a variable.

Q: What is the first step in solving for a variable?

A: The first step in solving for a variable is to isolate the variable on one side of the equation. This means that we need to get rid of the terms that involve the variable, so that we are left with only the terms that involve the variable.

Q: How do I get rid of terms that involve the variable?

A: To get rid of terms that involve the variable, we need to use inverse operations. For example, if we have a term that involves addition, we can use subtraction to get rid of it. If we have a term that involves multiplication, we can use division to get rid of it.

Q: What is the difference between an equation and an expression?

A: An equation is a statement that says two things are equal, while an expression is a group of numbers and variables that are combined using mathematical operations. For example, 2x + 3 is an expression, while 2x + 3 = 5 is an equation.

Q: How do I know which variable to solve for?

A: When solving for a variable, we need to identify the variable that we want to solve for. This is usually the variable that is isolated on one side of the equation. In the equation 6f + 9g = 3g + f, we want to solve for f, so we will isolate f on one side of the equation.

Q: What is the final step in solving for a variable?

A: The final step in solving for a variable is to check our work by plugging the solution back into the original equation. This ensures that our solution is correct and that we have isolated the variable correctly.

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 on one side of the equation
Not using inverse operations to get rid of terms that involve the variable
Not checking our work by plugging the solution back into the original equation

Q: How do I know if I have solved for the variable correctly?

A: To know if you have solved for the variable correctly, you need to check your work by plugging the solution back into the original equation. If the solution satisfies the equation, then you have solved for the variable correctly.

In conclusion, solving for a variable is a crucial skill in mathematics. By following the steps outlined above and avoiding common mistakes, we can isolate the variable and find its value. This skill is essential in mathematics, as it allows us to solve a wide range of problems involving linear equations.

Always isolate the variable on one side of the equation.
Use inverse operations to get rid of terms that involve the variable.
Check your work by plugging the solution back into the original equation.
Be careful when subtracting or adding terms to both sides of the equation.

Solve for x in the equation 2x + 5 = 3x - 2.
Solve for y in the equation 4y - 3 = 2y + 1.
Solve for z in the equation 3z + 2 = z - 4.

Solve for f in the equation 4f + 2g = 2f + 3g.
Solve for x in the equation 3x - 2 = 2x + 1.
Solve for y in the equation 2y + 3 = 4y - 2.