solve\:for\:f\:\left(a-\frac{b}{c}\right)\left(\frac{d}{e}+f\right)=g

To solve the equation for \( f \):

\[
\left(a - \frac{b}{c}\right)\left(\frac{d}{e} + f\right) = g
\]

### Step 1: Expand the Equation
First, distribute the term \( \left(a - \frac{b}{c}\right) \) across the terms inside the parentheses:

\[
\left(a - \frac{b}{c}\right) \cdot \frac{d}{e} + \left(a - \frac{b}{c}\right) \cdot f = g
\]

### Step 2: Simplify the Expression
Rewrite the equation:

\[
\frac{ad}{e} - \frac{bd}{ce} + \left(a - \frac{b}{c}\right)f = g
\]

### Step 3: Isolate the \( f \) Term
Subtract the non-\( f \) terms from both sides:

\[
\left(a - \frac{b}{c}\right)f = g - \frac{ad}{e} + \frac{bd}{ce}
\]

### Step 4: Solve for \( f \)
Finally, divide both sides by \( \left(a - \frac{b}{c}\right) \):

\[
f = \frac{g - \frac{ad}{e} + \frac{bd}{ce}}{a - \frac{b}{c}}
\]

This is the solution for \( f \).

Would you like any more details or have any questions?

Here are 5 related questions to explore further:

1. How would the solution change if \( g = 0 \)?
2. What happens if \( c = 1 \) in the original equation?
3. Can you solve for \( a \) instead of \( f \)?
4. What if \( \frac{d}{e} \) is a constant? How does it simplify the equation?
5. How would you approach the problem if \( a = b \) and \( d = e \)?

**Tip:** Always check if terms can be factored or simplified before solving equations, as it often makes the process easier.

Learn how to solve the equation (a - b/c)(d/e + f) = g step-by-step. This problem involves algebraic manipulation and distribution of terms. Suitable for advanced high school students.

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