To solve the equation $\left(\frac{\sum xy - b \sum x}{\sum x}\right) = \sum y - b \sum 1$, let's proceed step by step.

Step 1: Start with the given equation

$\frac{\sum xy - b \sum x}{\sum x} = \sum y - b \sum 1$

Step 2: Multiply both sides by $\sum x$ to eliminate the denominator

$\sum xy - b \sum x = \left(\sum y - b \sum 1\right) \sum x$

Step 3: Expand the right-hand side

$\sum xy - b \sum x = \sum y \sum x - b (\sum 1) (\sum x)$

Step 4: Rearrange terms to isolate the terms involving $b$

$\sum xy - \sum y \sum x = b \left(\sum x \sum 1 - \sum x\right)$

Step 5: Simplify the equation

Recognize that $\sum 1$ is simply the count of terms, denoted as $n$, so we have: $\sum x \sum 1 = n \sum x$ Thus, the equation becomes: $\sum xy - \sum y \sum x = b \left(n \sum x - \sum x\right)$

Step 6: Factor the right-hand side

$\sum xy - \sum y \sum x = b (n - 1) \sum x$

Step 7: Solve for $b$

$b = \frac{\sum xy - \sum y \sum x}{(n - 1) \sum x}$

This equation gives the value of $b$ in terms of the summations involving $x$ and $y$.

Would you like to dive into more details or have any further questions?

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Here are five related questions:

1. How do the sums $\sum x$ and $\sum y$ relate to mean values?
2. What are some scenarios where this equation might be useful?
3. How would the equation change if one of the sums were zero?
4. Can we generalize this approach for a different variable or function?
5. What role does the term $n - 1$ play in the context of statistical measures?

Tip: Always double-check your summation indices and limits to ensure the accuracy of your solutions, especially in complex equations.