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Solve this This is a problem from partial fractions 

The given expression is:

\[
\frac{3x - 1}{(x - 2)(9x + 3)}
\]

To decompose this into partial fractions, we express it as:

\[
\frac{3x - 1}{(x - 2)(9x + 3)} = \frac{A}{x - 2} + \frac{B}{9x + 3}
\]

### Step 1: Set Up the Equation

Multiply both sides by the denominator \((x - 2)(9x + 3)\) to eliminate the fraction:

\[
3x - 1 = A(9x + 3) + B(x - 2)
\]

### Step 2: Expand and Collect Like Terms

Expand the right side:

\[
3x - 1 = A(9x + 3) + B(x - 2)
\]

\[
3x - 1 = 9Ax + 3A + Bx - 2B
\]

Now, combine like terms:

\[
3x - 1 = (9A + B)x + (3A - 2B)
\]

### Step 3: Match Coefficients

Now, we match the coefficients of like terms from both sides:

For the coefficient of \(x\):

\[
9A + B = 3
\]

For the constant term:

\[
3A - 2B = -1
\]

### Step 4: Solve the System of Equations

Now, we solve the system of equations:

1. \(9A + B = 3\)
2. \(3A - 2B = -1\)

**From the first equation:**

\[
B = 3 - 9A
\]

**Substitute \(B = 3 - 9A\) into the second equation:**

\[
3A - 2(3 - 9A) = -1
\]

\[
3A - 6 + 18A = -1
\]

\[
21A - 6 = -1
\]

\[
21A = 5 \quad \Rightarrow \quad A = \frac{5}{21}
\]

**Substitute \(A = \frac{5}{21}\) back into \(B = 3 - 9A\):**

\[
B = 3 - 9\left(\frac{5}{21}\right) = 3 - \frac{45}{21} = 3 - \frac{15}{7} = \frac{21}{7} - \frac{15}{7} = \frac{6}{7}
\]

### Final Partial Fraction Decomposition:

Thus, the partial fraction decomposition is:

\[
\frac{3x - 1}{(x - 2)(9x + 3)} = \frac{\frac{5}{21}}{x - 2} + \frac{\frac{6}{7}}{9x + 3}
\]

Or, simplifying further:

\[
\frac{3x - 1}{(x - 2)(9x + 3)} = \frac{5}{21(x - 2)} + \frac{2}{7(3x + 1)}
\]

Would you like more details or have any questions about this solution?

### Here are some related questions:
1. How do you apply partial fraction decomposition to irreducible quadratic denominators?
2. What is the method of partial fractions used for in calculus?
3. How would you handle repeated linear factors in the denominator for partial fractions?
4. Can partial fraction decomposition be applied to improper fractions?
5. How does partial fraction decomposition aid in integrating rational functions?

**Tip:** When solving partial fractions, always double-check your algebraic manipulations and ensure the coefficients match correctly for accuracy.

Explore how to solve the problem from partial fractions using detailed step-by-step solutions. This problem covers partial fraction decomposition, algebraic manipulation, and matching coefficients. Suitable for college-level students and anyone interested in advanced algebraic techniques.

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