Solve for [tex]f[/tex]: [tex]6 f+9 g=3 g+f[/tex]

Simplify the given equation 6 f + 9 g = 3 g + f .
Isolate f on one side of the equation to get f = 5 − 6 g ​ .
Compare the simplified equation with the given options.
The correct solution is f = 5 − 6 g ​ , which matches option 2. f = 5 − 6 g ​ ​

Explanation

Problem Analysis We are given the equation 6 f + 9 g = 3 g + f and four possible solutions for f in terms of g . Our goal is to find which of the given solutions satisfies the equation.

Simplifying the Equation First, let's simplify the given equation:


6 f + 9 g = 3 g + f
Subtract f from both sides:
5 f + 9 g = 3 g
Subtract 9 g from both sides:
5 f = 3 g − 9 g
5 f = − 6 g
Divide both sides by 5:
f = 5 − 6 g ​

Comparing with Given Solutions Now, let's compare our simplified equation with the given possible solutions:

f = 3 − 8 g ​

f = 5 − 6 g ​

f = 6 − 5 g ​

f = 7 12 g ​


We can see that the second option, f = 5 − 6 g ​ , matches our simplified equation.

Final Answer Therefore, the correct solution is f = 5 − 6 g ​ .

Examples
Imagine you're balancing a seesaw. On one side, you have 6 friends and 9 watermelons. On the other side, you have 3 watermelons and 1 friend. To balance the seesaw, you need to figure out how many watermelons each friend is equivalent to. This problem is similar to solving the equation, where 'f' represents the number of friends and 'g' represents the number of watermelons. By solving the equation, you find the exact ratio of watermelons to friends needed to keep the seesaw balanced.

Answered by GinnyAnswer | 2025-07-05

The solution for f in the given equation 6 f + 9 g = 3 g + f is f = 5 − 6 g ​ . This equation was simplified step by step to isolate f . Therefore, the answer is f = 5 − 6 g ​ .
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Answered by Anonymous | 2025-07-19