Embark on a mathematical adventure as we delve into the intriguing equation “solve for f 6f 9g 3g f.” Join us as we decipher its complexities, unravel its secrets, and emerge with a profound understanding of its intricacies.
Our journey begins with an exploration of the mathematical operations that lie at the heart of this equation. We’ll break it down step by step, guiding you through the process of isolating the elusive variable “f.” Along the way, we’ll encounter algebraic techniques, variable isolation, and mathematical properties, each playing a crucial role in our quest for a solution.
Equation Analysis
The equation “solve for f 6f 9g 3g f” involves a combination of mathematical operations, including multiplication and addition.
To isolate the variable “f” and solve for its value, we can follow these steps:
Simplifying the Equation
First, combine like terms on both sides of the equation:
- 6f + f = 9g + 3g
- 7f = 12g
Isolating the Variable
To isolate “f,” we need to divide both sides of the equation by 7:
- (7f) / 7 = (12g) / 7
- f = (12g) / 7
Examples
Here are a few examples to illustrate the process:
- If g = 3, then f = (12 – 3) / 7 = 5.14
- If g = 5, then f = (12 – 5) / 7 = 8.57
- If g = 7, then f = (12 – 7) / 7 = 12
Algebraic Techniques
Algebraic techniques are essential tools for solving equations. These techniques simplify complex equations, making them easier to understand and solve. Two fundamental algebraic techniques are combining like terms and factoring.
Combining Like Terms
Combining like terms involves adding or subtracting terms that have the same variable and exponent. For instance, in the equation 6f + 9g + 3g – f, the terms 6f and -f can be combined to give 5f. Similarly, 9g and 3g can be combined to give 12g.
Factoring
Factoring involves expressing an algebraic expression as a product of simpler factors. For example, in the equation 6f + 9g + 3g – f, the left-hand side can be factored as 5f + 12g. Factoring can make it easier to solve equations by isolating the variables and reducing the number of terms.
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Variable Isolation
Isolating a variable means getting it by itself on one side of the equation. This is crucial for solving the equation because it allows us to find the value of the variable that makes the equation true.
Steps to Isolate “f”
- Subtract 9g from both sides:This removes 9g from the left side and the right side, leaving us with 6f = 3g.
- Divide both sides by 6:This divides both sides by 6, giving us f = 3g/6.
- Simplify:We can simplify 3g/6 to 1/2g, so the final solution is f = 1/2g.
By following these steps, we have successfully isolated “f” on the left side of the equation and expressed it in terms of “g”.
Mathematical Properties: Solve For F 6f 9g 3g F
Mathematical properties are essential tools for solving equations, as they allow us to manipulate the equation without altering its solution. In solving 6f + 9g = 3g + f, we utilize two key properties: the distributive property and the additive inverse property.
The distributive property states that a(b + c) = ab + ac. This property allows us to distribute a coefficient across a sum or difference within parentheses. In our equation, we can apply the distributive property to the left-hand side (LHS) as follows:
Distributive Property, Solve for f 6f 9g 3g f
- 6f + 9g = 3g + f
- 6f + (9g – f) = 3g
- 6f + 8g = 3g
By distributing the coefficient 6 across the parentheses, we simplify the LHS and make it easier to solve for f.
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The additive inverse property states that for any number a, there exists an additive inverse -a such that a + (-a) = 0. This property allows us to add the additive inverse of a term to both sides of an equation without changing its solution.
Additive Inverse Property
- 6f + 8g = 3g
- 6f + 8g – 3g = 3g – 3g
- 6f + 5g = 0
By adding the additive inverse of 3g to both sides of the equation, we eliminate the term 3g and simplify the equation further.
Table of Solutions
To demonstrate the process of solving the equation for different values of “g,” we can create a table with four columns: “Equation,” “Step 1,” “Step 2,” and “Solution.”
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Populating the Table
The table below shows the equation, the first step in solving for “f,” the second step, and the final solution for different values of “g.”
| Equation | Step 1 | Step 2 | Solution |
|---|---|---|---|
| 6f + 9g = 3g + f | Subtract f from both sides: 5f + 9g = 3g | Subtract 3g from both sides: 5f = 6g | f = 6g/5 |
| 6f + 9g = 3g + 2f | Subtract 2f from both sides: 4f + 9g = 3g | Subtract 3g from both sides: 4f = 6g | f = 6g/4 |
| 6f + 9g = 3g + 4f | Subtract 4f from both sides: 2f + 9g = 3g | Subtract 3g from both sides: 2f = 6g | f = 6g/2 = 3g |
Visual Representation
To visually represent the steps involved in solving the equation for “f,” a flowchart can be designed. This flowchart will use arrows and labels to illustrate the flow of the solution process and employ colors and shapes to make it visually appealing and easy to understand.
Flowchart
The flowchart will have the following steps:
- Start:Given the equation 6f + 9g = 3g + f.
- Subtract 3g from both sides:6f + 9g
- 3g = 3g + f
- 3g.
- Simplify:6f + 6g = f.
- Subtract f from both sides:6f + 6g
- f = f
- f.
- Simplify:5f + 6g = 0.
- Solve for f:f = (-6/5)g.
- End:The solution for “f” is f = (-6/5)g.
The flowchart will use the following colors and shapes:
- Start and End:Green circles.
- Steps:Blue rectangles.
- Arrows:Black arrows.
The flowchart will be visually appealing and easy to understand, providing a clear representation of the steps involved in solving the equation for “f.”
Top FAQs
What is the significance of isolating the variable “f” in the equation?
Isolating “f” is essential because it allows us to determine its value independently of the other variables in the equation.
How do algebraic techniques simplify the equation?
Algebraic techniques, such as combining like terms and factoring, enable us to manipulate the equation into a simpler form, making it easier to solve.
What is the role of mathematical properties in solving the equation?
Mathematical properties, like the distributive and additive inverse properties, provide rules that allow us to transform the equation without altering its solution.
