To solve the equation

$\frac{\left(\sum x \cdot y\right) - \left(b \sum x\right)}{\sum x} = \sum y - b \cdot \sum 1,$

we will break it down step by step.

Step 1: Simplify the Left-Hand Side (LHS)

The LHS is:

$\frac{\left(\sum x \cdot y\right) - \left(b \sum x\right)}{\sum x}$

This can be rewritten as:

$\frac{\sum x \cdot y}{\sum x} - \frac{b \sum x}{\sum x}$

Simplify each term:

$\frac{\sum x \cdot y}{\sum x} - b$

Step 2: Set the LHS Equal to the Right-Hand Side (RHS)

Now, let's set this equal to the RHS, which is $\sum y - b \sum 1$:

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

Step 3: Isolate the term involving $b$

To isolate $b$, add $b$ to both sides:

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

Now, factor $b$ out on the right-hand side:

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

Step 4: Solve for $b$

Isolate $b$ by moving the other terms:

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

Thus, solving for $b$ gives:

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

Final Solution

The final expression for $b$ is:

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

This is the value of $b$ in terms of the sums provided.

Would you like more details or have any questions about this solution? Here are five related questions to consider:

1. How can the equation be applied to a specific data set?
2. What is the geometric interpretation of this equation?
3. How does this relate to the equation of a line in linear regression?
4. What assumptions are necessary for the sums $\sum x$ and $\sum 1$?
5. How does changing the value of $x$ or $y$ affect the value of $b$?

Tip: Understanding the relationships between sums and their components can help in deriving expressions like this more intuitively.