Answer
The equation of the least squares line associated with the given set of data points is $y=\frac{-11}{114}x+\frac{7}{5}$
Work Step by Step
The matrices can be formed as:
$A=\begin{bmatrix}
-7& 1 \\
-4& 1 \\
2 & 1\\
3& 1\\
6 & 1
\end{bmatrix} \rightarrow A^T=\begin{bmatrix}
-7& -4&2&3&6 \\
1 & 1&1 &1&1\\\end{bmatrix}$
$x=\begin{bmatrix}
a \\
b \\
\end{bmatrix}$
$b=\begin{bmatrix}
3\\
0\\
-1 \\
6\\
-1
\end{bmatrix}$
Apply matrices to the least square solution:
$x_0=(A^TA)^{-1}A^Tb$
$=(\begin{bmatrix}
-7& -4&2&3&6 \\
1 & 1&1 &1&1\\\end{bmatrix}\begin{bmatrix}
-7& 1 \\
-4& 1 \\
2 & 1\\
3& 1\\
6 & 1
\end{bmatrix})^{-1}\begin{bmatrix}
-7& -4&2&3&6 \\
1 & 1&1 &1&1\\\end{bmatrix} \begin{bmatrix}
3\\
0\\
-1 \\
6\\
-1
\end{bmatrix}$
$=\begin{bmatrix}
114 &0\\
0&5
\end{bmatrix}^{-1}\begin{bmatrix}
-7& -4&2&3&6 \\
1 & 1&1 &1&1\\\end{bmatrix}\begin{bmatrix}
3\\
0\\
-1 \\
6\\
-1
\end{bmatrix}$
$=\frac{1}{570}\begin{bmatrix}
5 & 0\\
0& 114
\end{bmatrix}\begin{bmatrix}
-7& -4&2&3&6 \\
1 & 1&1 &1&1\\\end{bmatrix}\begin{bmatrix}
3\\
0\\
-1 \\
6\\
-1
\end{bmatrix}$
$=\frac{1}{570}\begin{bmatrix}
-35 &-20&10&15&30 \\
114& 114 &114&114&114
\end{bmatrix}\begin{bmatrix}
3\\
0\\
-1 \\
6\\
-1
\end{bmatrix}$
$=\frac{1}{570}\begin{bmatrix}
-55 \\
798
\end{bmatrix}$
$=\begin{bmatrix}
\frac{-11}{114} \\
\frac{7}{5}
\end{bmatrix}$
The equation of the least squares line associated with the given set of data points is $y=\frac{-11}{114}x+\frac{7}{5}$
