Answer
The equation of the least squares line associated with the given set of data points is $y=\frac{40}{3}$
Work Step by Step
The matrices can be formed as:
$A=\begin{bmatrix}
1& 1 \\
2& 1 \\
3 & 1\\
\end{bmatrix} \rightarrow A^T=\begin{bmatrix}
1& 2&3 \\
1 & 1&1 \\\end{bmatrix}$
$x=\begin{bmatrix}
a \\
b \\
\end{bmatrix}$
$b=\begin{bmatrix}
10\\
20\\
10 \\
\end{bmatrix}$
Apply matrices to the least square solution:
$x_0=(A^TA)^{-1}A^Tb$
$=(\begin{bmatrix}
1& 2&3 \\
1 & 1&1 \\\end{bmatrix}\begin{bmatrix}
1& 1 \\
2& 1 \\
3 & 1\\
\end{bmatrix} )^{-1}\begin{bmatrix}
1& 2&3 \\
1 & 1&1 \\\end{bmatrix} \begin{bmatrix}
10\\
20\\
10 \\
\end{bmatrix}$
$=\begin{bmatrix}
14 &6\\
6&3
\end{bmatrix}^{-1}\begin{bmatrix}
1& 2&3 \\
1 & 1&1 \\\end{bmatrix}\begin{bmatrix}
10\\
20\\
10 \\
\end{bmatrix}$
$=\frac{1}{6}\begin{bmatrix}
3 & -6\\
-6& 14
\end{bmatrix}\begin{bmatrix}
1& 2&3 \\
1 & 1&1 \\\end{bmatrix}\begin{bmatrix}
10\\
20\\
10 \\
\end{bmatrix}$
$=\frac{1}{6}\begin{bmatrix}
-3 &0&3 \\
8& 2&-4
\end{bmatrix}\begin{bmatrix}
10\\
20\\
10 \\
\end{bmatrix}$
$=\frac{1}{6}\begin{bmatrix}
0 \\
80\end{bmatrix}$
$=\begin{bmatrix}
0 \\
\frac{40}{3}
\end{bmatrix}$
The equation of the least squares line associated with the given set of data points is $y=\frac{40}{3}$
