To solve the given equation for $b$, let's start by clearly rewriting the equation:

$\frac{(\sum x \cdot y) - b \sum x}{(\sum x)^2} = \frac{(\sum y) - b \sum 1}{\sum x}$

Next, we can follow these steps:

1. Cross-multiply to eliminate the fractions:

$\left((\sum x \cdot y) - b \sum x\right) \cdot \sum x = \left((\sum y) - b \sum 1\right) \cdot (\sum x)^2$

2. Expand both sides:

$(\sum x \cdot y) \cdot \sum x - b (\sum x) \cdot \sum x = (\sum y) \cdot (\sum x)^2 - b (\sum 1) \cdot (\sum x)^2$

3. Group the terms involving $b$ on one side of the equation:

$b (\sum x) \cdot \sum x - b (\sum 1) \cdot (\sum x)^2 = (\sum y) \cdot (\sum x)^2 - (\sum x \cdot y) \cdot \sum x$

4. Factor out $b$ on the left-hand side:

$b \left[(\sum x)^2 - (\sum 1) \cdot (\sum x)^2\right] = (\sum y) \cdot (\sum x)^2 - (\sum x \cdot y) \cdot \sum x$

5. Simplify the left-hand side by factoring:

$b \left[(\sum x)^2 \left(1 - \sum 1\right)\right] = (\sum y) \cdot (\sum x)^2 - (\sum x \cdot y) \cdot \sum x$

6. Isolate $b$:

$b = \frac{(\sum y) \cdot (\sum x)^2 - (\sum x \cdot y) \cdot \sum x}{(\sum x)^2 \cdot (1 - \sum 1)}$

Thus, the solution for $b$ is:

$\boxed{b = \frac{(\sum y) \cdot (\sum x)^2 - (\sum x \cdot y) \cdot \sum x}{(\sum x)^2 \cdot (1 - \sum 1)}}$

Note: If $\sum 1 = 1$, the equation is undefined, which implies that the problem may have been misinterpreted or that the assumptions about the terms need to be reconsidered.

Would you like more details on any of the steps?

Here are some questions to consider:

1. How would the solution change if the assumption about $\sum 1$ is different?
2. What might cause the equation to become undefined, and how can this be avoided?
3. Can this solution be simplified under certain conditions?
4. How does this equation relate to the slope calculation in a linear regression?
5. What assumptions are necessary for the solution to be valid in a practical scenario?

Tip: Always be cautious of potential undefined conditions when dealing with equations that involve division.