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We are asked to resolve the expression \(\frac{7x - 1}{6x^2 - 5x + 1}\) into partial fractions.

### Step 1: Factor the denominator
We need to factor the quadratic expression \(6x^2 - 5x + 1\).

To factor \(6x^2 - 5x + 1\), look for two numbers that multiply to \(6 \times 1 = 6\) and add to \(-5\). These numbers are \(-3\) and \(-2\). So, we can rewrite the quadratic as:
\[
6x^2 - 5x + 1 = (2x - 1)(3x - 1)
\]

### Step 2: Set up the partial fractions
Now that the denominator is factored, we can express the fraction as the sum of two partial fractions:
\[
\frac{7x - 1}{(2x - 1)(3x - 1)} = \frac{A}{2x - 1} + \frac{B}{3x - 1}
\]
where \(A\) and \(B\) are constants to be determined.

### Step 3: Solve for \(A\) and \(B\)
Multiply both sides by \((2x - 1)(3x - 1)\) to eliminate the denominators:
\[
7x - 1 = A(3x - 1) + B(2x - 1)
\]
Expand both sides:
\[
7x - 1 = A(3x - 1) + B(2x - 1)
\]
\[
7x - 1 = A(3x) - A + B(2x) - B
\]
\[
7x - 1 = (3A + 2B)x - (A + B)
\]
Now, equate the coefficients of \(x\) and the constant terms:
\[
3A + 2B = 7 \quad \text{(coefficient of \(x\))}
\]
\[
-(A + B) = -1 \quad \text{(constant term)}
\]
From the second equation, we have:
\[
A + B = 1
\]
Substitute \(A = 1 - B\) into the first equation:
\[
3(1 - B) + 2B = 7
\]
\[
3 - 3B + 2B = 7
\]
\[
3 - B = 7
\]
\[
B = -4
\]
Now substitute \(B = -4\) into \(A + B = 1\):
\[
A - 4 = 1
\]
\[
A = 5
\]

### Step 4: Final answer
Substitute the values of \(A = 5\) and \(B = -4\) into the partial fraction expression:
\[
\frac{7x - 1}{(2x - 1)(3x - 1)} = \frac{5}{2x - 1} - \frac{4}{3x - 1}
\]

This is the decomposition into partial fractions.

Would you like more details or have any questions? 

Here are some related questions to explore:
1. How do we factor quadratic expressions in general?
2. What are the methods to solve systems of linear equations like in this case?
3. Can partial fraction decomposition be applied to higher-degree polynomials?
4. How can partial fractions be used to solve integrals?
5. How does the method differ when the denominator has repeated roots?

**Tip:** Always ensure the denominator is factorable before proceeding with partial fraction decomposition.

Resolve (7x - 1) / (6x^2 - 5x + 1) into partial fractions.

Learn how to resolve (7x - 1) / (6x^2 - 5x + 1) into partial fractions through factoring and algebraic methods. This step-by-step solution explains the key concepts of partial fraction decomposition and factoring quadratic equations, suitable for Grades 11-12 and college students.

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