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

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

We can proceed by following 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:

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

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. Isolate $b$:

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

So the solution for $b$ is:

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

Would you like more details on any of the steps?

Here are some questions that could help deepen understanding:

1. What is the significance of each summation term in this equation?
2. How does this equation relate to concepts in statistics, such as linear regression?
3. Can this form of the equation be simplified further if certain assumptions about the data are made?
4. How would the equation change if $\sum 1$ equals the number of terms $n$?
5. How does this formula compare with the general formula for the slope in linear regression?

Tip: Cross-multiplying is a helpful technique to eliminate fractions and simplify equations involving ratios.