The Least Squares Line The least squares line or regression line is the line that best fits a set of points in the plane. We studied this line in the Focus on Modeling that follows Chapter 2 (see page $171 ) .$ By using calculus, it can be shown that the line that best fits the $n$ data points $\left(x_{1}, = y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{m} y_{n}\right)$ is the line $y=a x+b,$ where the coefficients $a$ and $b$ satisfy the following pair of linear equations. (The notation $\sum_{k=1}^{n} x_{k}$ stands for the sum of all the $X^{\prime}$ see Section 13.1 for a complete description of sigma ( $\Sigma$ ) notation.
$$\left(\sum_{k=1}^{n} x_{k}\right) a+n b=\sum_{k=1}^{n} y_{k}$$
$$\left(\sum_{k=1}^{n} x_{k}^{2}\right) a+\left(\sum_{k=1}^{n} x_{k}\right) b=\sum_{k=1}^{n} x_{k} y_{k}$$
Use these equations to find the least squares line for the following data points.
$$(1,3), \quad(2,5), \quad(3,6), \quad(5,6), \quad(7,9)$$
Sketch the points and your line to confirm that the line fits these points well. If your calculator computes regression lines, see whether it gives you the same line as the formulas.