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!

Learn how to solve the algebraic rational equation b/a + c/b = 1/ab for the variable a. This solution includes step-by-step elimination of fractions and rearrangement of terms. Suitable for students in Grades 9-12 who are studying algebra and rational equations.

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