We are given the equation:

$\frac{b}{a} + \frac{c}{b} = \frac{1}{ab}$

Let's solve for $a$.

Step 1: Eliminate denominators

Multiply through the equation by $ab$ to eliminate the denominators. This gives:

$ab \cdot \left( \frac{b}{a} + \frac{c}{b} \right) = ab \cdot \frac{1}{ab}$

Step 2: Simplify each term

For the left-hand side:

$ab \cdot \frac{b}{a} = b^2, \quad ab \cdot \frac{c}{b} = ac$

For the right-hand side:

$ab \cdot \frac{1}{ab} = 1$

Now the equation becomes:

$b^2 + ac = 1$

Step 3: Solve for $a$

Rearrange the equation to solve for $a$:

$ac = 1 - b^2$

Now, divide both sides by $c$ to isolate $a$:

$a = \frac{1 - b^2}{c}$

Final Answer:

$a = \frac{1 - b^2}{c}$

Let me know if you want more details on the solution or have any other questions! Here are some related questions:

1. How would the solution change if the equation had a different constant instead of 1 on the right-hand side?
2. Can we solve for $b$ instead of $a$ in the same equation?
3. How would we handle this problem if $c = 0$?
4. What is the geometric interpretation of this equation if we assign physical meaning to $a$, $b$, and $c$?
5. How do we solve equations like this when they involve more complex terms, like higher powers or square roots?

Tip: Always check if you can simplify the equation by eliminating fractions early on—this makes solving much easier!