We are given the equation:

$\frac{x + 3}{10} = \frac{8}{5} + \frac{x - 6}{7}$

Step 1: Eliminate the fractions by finding a common denominator

The denominators are 10, 5, and 7. The least common denominator (LCD) of 10, 5, and 7 is 70.

Multiply both sides of the equation by 70 to eliminate the fractions:

$70 \times \frac{x + 3}{10} = 70 \times \left(\frac{8}{5} + \frac{x - 6}{7}\right)$

This simplifies as follows:

$7(x + 3) = 14(8) + 10(x - 6)$

Step 2: Simplify both sides

Now, simplify each part:

$7(x + 3) = 14 \times 8 + 10(x - 6)$

$7x + 21 = 112 + 10x - 60$

Simplifying further:

$7x + 21 = 10x + 52$

Step 3: Solve for $x$

To solve for $x$, first move all terms involving $x$ to one side and constant terms to the other side. Subtract $7x$ from both sides:

$21 = 3x + 52$

Now, subtract 52 from both sides:

$-31 = 3x$

Finally, divide by 3:

$x = \frac{-31}{3}$

Step 4: Check the solution

We substitute $x = \frac{-31}{3}$ into the original equation to verify if both sides are equal. This involves plugging the value into both fractions and checking if they match.

Thus, the solution is:

$\boxed{x = \frac{-31}{3}}$

The correct answer is A. The solution set is $\left\{ \frac{-31}{3} \right\}$.

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Would you like more details on how to check this solution or have any further questions?

Here are 5 related questions:

1. How do you find the least common denominator (LCD) for fractions?
2. What steps are involved in eliminating fractions from an equation?
3. Can you solve a similar equation with three fractions instead of two?
4. What strategies can help simplify complex linear equations faster?
5. Why is it important to check the solution after solving an equation?

Tip: When dealing with equations containing fractions, clearing the denominators by multiplying both sides by the least common denominator is a highly effective first step.