To solve the equation for \(a\), we start with the given equation:

\[
\mathbf{u} = \frac{K}{a(w + v)}
\]

To isolate \(a\), we follow these steps:

1. **Multiply both sides by \(a(w + v)\):**
   \[
   \mathbf{u} \cdot a(w + v) = K
   \]

2. **Divide both sides by \(\mathbf{u} \cdot (w + v)\):**
   \[
   a = \frac{K}{\mathbf{u} \cdot (w + v)}
   \]

So the formula for \(a\) is:

\[
a = \frac{K}{\mathbf{u} \cdot (w + v)}
\]

This expression now gives you \(a\) in terms of \(K\), \(\mathbf{u}\), \(w\), and \(v\).

Would you like further clarification or details on this solution?

Here are some additional questions that may help you explore this further:

1. How does the value of \(a\) change if \(K\) increases?
2. What happens to \(a\) if \(\mathbf{u}\) increases while other variables remain constant?
3. How does \(w + v\) influence the value of \(a\)?
4. If \(w\) and \(v\) represent velocities, what physical interpretation could \(a\) have?
5. Can you substitute specific values into the formula to calculate \(a\)?
6. What happens to \(a\) if \(K\) is held constant, but \(w\) and \(v\) are changed?
7. Is there a real-world situation where this formula for \(a\) might be directly applicable?
8. How would you interpret the case where \(K = 0\) in this context?

**Tip:** When solving equations for a specific variable, always double-check your algebraic manipulations to ensure accuracy.

Learn how to solve the equation u = K / (a(w + v)) to find the value of variable a. This problem involves algebraic manipulations and isolating variables, suitable for advanced high school students (Grades 10-12).

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