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### (a) Finding the least squares regression line \( y = ax + b \)

We have the system of equations for finding the coefficients \(a\) (slope) and \(b\) (intercept):

\[
\begin{align*}
3.00b + 3.70a &= 108.00 \\
3.70b + 4.69a &= 127.70
\end{align*}
\]

This system can be solved using algebra. Let's solve for \(a\) and \(b\).The solution to the system of equations gives:

- Slope \( a = -43.42 \) (rounded to two decimal places)
- Intercept \( b = 89.55 \) (rounded to two decimal places)

Thus, the least squares regression line is:

\[
y = -43.42x + 89.55
\]

Would you like more details on how this was derived or any clarification?

Here are five related questions for you to consider:
1. What does the negative slope indicate about the relationship between price and demand?
2. How could the demand be predicted for a new price, say \( x = 1.75 \)?
3. What is the meaning of the intercept in the context of this problem?
4. How would the model change if a new data point were added?
5. What are the assumptions behind using least squares regression?

**Tip:** Remember that a negative slope implies an inverse relationship—meaning as price increases, demand decreases.

A candy store manager wants to know the demand for a candy bar as a function of the price. The table shows the daily sales for different prices, and the goal is to find the least squares regression line y = ax + b for the data, by solving the system for a and b.

Learn how to find the least squares regression line for a candy store's price and demand data using the system of equations. This problem demonstrates the application of algebra and the least squares regression theorem. Suitable for students in Grades 10-12.

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