Mister Exam
System of equations solver/
a+b+c=0; -a*((1+sqrt(65))/8)-b*((1-sqrt(65))/8)+c+d=23; a+b-c+d=-2; -a((1+sqrt(65))/8)-b((1-sqrt(65))/8)-d=11
a+b+c=0; -a*((1+sqrt(65))/8)-b*((1-sqrt(65))/8)+c+d=23; a+b-c+d=-2; -a((1+sqrt(65))/8)-b((1-sqrt(65))/8)-d=11
The solution
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a + b + c = 0
$$c + \left(a + b\right) = 0$$
____ ____
1 + \/ 65 1 - \/ 65
-a*---------- - b*---------- + c + d = 23
8 8
$$d + \left(c + \left(- a \frac{1 + \sqrt{65}}{8} - b \frac{1 - \sqrt{65}}{8}\right)\right) = 23$$
a + b - c + d = -2
$$d + \left(- c + \left(a + b\right)\right) = -2$$
____ ____
1 + \/ 65 1 - \/ 65
-a*---------- - b*---------- - d = 11
8 8
$$- d + \left(- a \frac{1 + \sqrt{65}}{8} - b \frac{1 - \sqrt{65}}{8}\right) = 11$$
-d + (-a)*((1 + sqrt(65))/8) - b*(1 - sqrt(65))/8 = 11
Rapid solution
$$a_{1} = - \frac{12 \sqrt{65}}{13} - \frac{8}{5}$$
=
$$- \frac{12 \sqrt{65}}{13} - \frac{8}{5}$$
=
$$b_{1} = - \frac{8}{5} + \frac{12 \sqrt{65}}{13}$$
=
$$- \frac{8}{5} + \frac{12 \sqrt{65}}{13}$$
=
$$c_{1} = \frac{16}{5}$$
=
$$\frac{16}{5}$$
=
$$d_{1} = \frac{22}{5}$$
=
$$\frac{22}{5}$$
=
=
$$- \frac{12 \sqrt{65}}{13} - \frac{8}{5}$$
=
-9.04208407535251
$$b_{1} = - \frac{8}{5} + \frac{12 \sqrt{65}}{13}$$
=
$$- \frac{8}{5} + \frac{12 \sqrt{65}}{13}$$
=
5.84208407535251
$$c_{1} = \frac{16}{5}$$
=
$$\frac{16}{5}$$
=
3.2
$$d_{1} = \frac{22}{5}$$
=
$$\frac{22}{5}$$
=
4.4
Gaussian elimination
Given the system of equations
$$c + \left(a + b\right) = 0$$
$$d + \left(c + \left(- a \frac{1 + \sqrt{65}}{8} - b \frac{1 - \sqrt{65}}{8}\right)\right) = 23$$
$$d + \left(- c + \left(a + b\right)\right) = -2$$
$$- d + \left(- a \frac{1 + \sqrt{65}}{8} - b \frac{1 - \sqrt{65}}{8}\right) = 11$$
We give the system of equations to the canonical form
$$a + b + c = 0$$
$$- \frac{\sqrt{65} a}{8} - \frac{a}{8} - \frac{b}{8} + \frac{\sqrt{65} b}{8} + c + d - 23 = 0$$
$$a + b - c + d = -2$$
$$- \frac{\sqrt{65} a}{8} - \frac{a}{8} - \frac{b}{8} + \frac{\sqrt{65} b}{8} - d - 11 = 0$$
Rewrite the system of linear equations as the matrix form
$$\left[\begin{matrix}1 & 1 & 1 & 0 & 0\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 1 & 1 & 23\\1 & 1 & -1 & 1 & -2\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 0 & -1 & 11\end{matrix}\right]$$
In 1 -th column
$$\left[\begin{matrix}1\\- \frac{\sqrt{65}}{8} - \frac{1}{8}\\1\\- \frac{\sqrt{65}}{8} - \frac{1}{8}\end{matrix}\right]$$
let’s convert all the elements, except
1 -th element into zero.
- To do this, let’s take 1 -th line
$$\left[\begin{matrix}1 & 1 & 1 & 0 & 0\end{matrix}\right]$$
,
and subtract it from other lines:
From 2 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}\left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right) - \left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right) & \left(- \frac{1}{8} + \frac{\sqrt{65}}{8}\right) - \left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right) & 1 - \left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right) & 1 - 0 \left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right) & 23 - 0 \left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right)\end{matrix}\right] = \left[\begin{matrix}0 & \frac{\sqrt{65}}{4} & \frac{\sqrt{65}}{8} + \frac{9}{8} & 1 & 23\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 1 & 1 & 0 & 0\\0 & \frac{\sqrt{65}}{4} & \frac{\sqrt{65}}{8} + \frac{9}{8} & 1 & 23\\1 & 1 & -1 & 1 & -2\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 0 & -1 & 11\end{matrix}\right]$$
From 3 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}-1 + 1 & -1 + 1 & -1 - 1 & \left(-1\right) 0 + 1 & -2 + \left(-1\right) 0\end{matrix}\right] = \left[\begin{matrix}0 & 0 & -2 & 1 & -2\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 1 & 1 & 0 & 0\\0 & \frac{\sqrt{65}}{4} & \frac{\sqrt{65}}{8} + \frac{9}{8} & 1 & 23\\0 & 0 & -2 & 1 & -2\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 0 & -1 & 11\end{matrix}\right]$$
From 4 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}\left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right) - \left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right) & \left(- \frac{1}{8} + \frac{\sqrt{65}}{8}\right) - \left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right) & - (- \frac{\sqrt{65}}{8} - \frac{1}{8}) & -1 - 0 \left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right) & 11 - 0 \left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right)\end{matrix}\right] = \left[\begin{matrix}0 & \frac{\sqrt{65}}{4} & \frac{1}{8} + \frac{\sqrt{65}}{8} & -1 & 11\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 1 & 1 & 0 & 0\\0 & \frac{\sqrt{65}}{4} & \frac{\sqrt{65}}{8} + \frac{9}{8} & 1 & 23\\0 & 0 & -2 & 1 & -2\\0 & \frac{\sqrt{65}}{4} & \frac{1}{8} + \frac{\sqrt{65}}{8} & -1 & 11\end{matrix}\right]$$
In 2 -th column
$$\left[\begin{matrix}1\\\frac{\sqrt{65}}{4}\\0\\\frac{\sqrt{65}}{4}\end{matrix}\right]$$
let’s convert all the elements, except
1 -th element into zero.
- To do this, let’s take 1 -th line
$$\left[\begin{matrix}1 & 1 & 1 & 0 & 0\end{matrix}\right]$$
,
and subtract it from other lines:
From 2 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- \frac{\sqrt{65}}{4} & - \frac{\sqrt{65}}{4} + \frac{\sqrt{65}}{4} & - \frac{\sqrt{65}}{4} + \left(\frac{\sqrt{65}}{8} + \frac{9}{8}\right) & 1 - 0 \frac{\sqrt{65}}{4} & 23 - 0 \frac{\sqrt{65}}{4}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{65}}{4} & 0 & \frac{9}{8} - \frac{\sqrt{65}}{8} & 1 & 23\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 1 & 1 & 0 & 0\\- \frac{\sqrt{65}}{4} & 0 & \frac{9}{8} - \frac{\sqrt{65}}{8} & 1 & 23\\0 & 0 & -2 & 1 & -2\\0 & \frac{\sqrt{65}}{4} & \frac{1}{8} + \frac{\sqrt{65}}{8} & -1 & 11\end{matrix}\right]$$
From 4 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- \frac{\sqrt{65}}{4} & - \frac{\sqrt{65}}{4} + \frac{\sqrt{65}}{4} & - \frac{\sqrt{65}}{4} + \left(\frac{1}{8} + \frac{\sqrt{65}}{8}\right) & -1 - 0 \frac{\sqrt{65}}{4} & 11 - 0 \frac{\sqrt{65}}{4}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{65}}{4} & 0 & \frac{1}{8} - \frac{\sqrt{65}}{8} & -1 & 11\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 1 & 1 & 0 & 0\\- \frac{\sqrt{65}}{4} & 0 & \frac{9}{8} - \frac{\sqrt{65}}{8} & 1 & 23\\0 & 0 & -2 & 1 & -2\\- \frac{\sqrt{65}}{4} & 0 & \frac{1}{8} - \frac{\sqrt{65}}{8} & -1 & 11\end{matrix}\right]$$
In 3 -th column
$$\left[\begin{matrix}1\\\frac{9}{8} - \frac{\sqrt{65}}{8}\\-2\\\frac{1}{8} - \frac{\sqrt{65}}{8}\end{matrix}\right]$$
let’s convert all the elements, except
3 -th element into zero.
- To do this, let’s take 3 -th line
$$\left[\begin{matrix}0 & 0 & -2 & 1 & -2\end{matrix}\right]$$
,
and subtract it from other lines:
From 1 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}1 - \frac{\left(-1\right) 0}{2} & 1 - \frac{\left(-1\right) 0}{2} & 1 - - -1 & - \frac{-1}{2} & - \frac{\left(-1\right) \left(-1\right) 2}{2}\end{matrix}\right] = \left[\begin{matrix}1 & 1 & 0 & \frac{1}{2} & -1\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 1 & 0 & \frac{1}{2} & -1\\- \frac{\sqrt{65}}{4} & 0 & \frac{9}{8} - \frac{\sqrt{65}}{8} & 1 & 23\\0 & 0 & -2 & 1 & -2\\- \frac{\sqrt{65}}{4} & 0 & \frac{1}{8} - \frac{\sqrt{65}}{8} & -1 & 11\end{matrix}\right]$$
From 2 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- \frac{\sqrt{65}}{4} - 0 \left(- \frac{9}{16} + \frac{\sqrt{65}}{16}\right) & - 0 \left(- \frac{9}{16} + \frac{\sqrt{65}}{16}\right) & - \left(-1\right) 2 \left(- \frac{9}{16} + \frac{\sqrt{65}}{16}\right) + \left(\frac{9}{8} - \frac{\sqrt{65}}{8}\right) & 1 - \left(- \frac{9}{16} + \frac{\sqrt{65}}{16}\right) & 23 - - 2 \left(- \frac{9}{16} + \frac{\sqrt{65}}{16}\right)\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{65}}{4} & 0 & 0 & \frac{25}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{175}{8}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 1 & 0 & \frac{1}{2} & -1\\- \frac{\sqrt{65}}{4} & 0 & 0 & \frac{25}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{175}{8}\\0 & 0 & -2 & 1 & -2\\- \frac{\sqrt{65}}{4} & 0 & \frac{1}{8} - \frac{\sqrt{65}}{8} & -1 & 11\end{matrix}\right]$$
From 4 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- \frac{\sqrt{65}}{4} - 0 \left(- \frac{1}{16} + \frac{\sqrt{65}}{16}\right) & - 0 \left(- \frac{1}{16} + \frac{\sqrt{65}}{16}\right) & \left(\frac{1}{8} - \frac{\sqrt{65}}{8}\right) - - 2 \left(- \frac{1}{16} + \frac{\sqrt{65}}{16}\right) & -1 - \left(- \frac{1}{16} + \frac{\sqrt{65}}{16}\right) & 11 - - 2 \left(- \frac{1}{16} + \frac{\sqrt{65}}{16}\right)\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{65}}{4} & 0 & 0 & - \frac{15}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{87}{8}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 1 & 0 & \frac{1}{2} & -1\\- \frac{\sqrt{65}}{4} & 0 & 0 & \frac{25}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{175}{8}\\0 & 0 & -2 & 1 & -2\\- \frac{\sqrt{65}}{4} & 0 & 0 & - \frac{15}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{87}{8}\end{matrix}\right]$$
In 4 -th column
$$\left[\begin{matrix}\frac{1}{2}\\\frac{25}{16} - \frac{\sqrt{65}}{16}\\1\\- \frac{15}{16} - \frac{\sqrt{65}}{16}\end{matrix}\right]$$
let’s convert all the elements, except
2 -th element into zero.
- To do this, let’s take 2 -th line
$$\left[\begin{matrix}- \frac{\sqrt{65}}{4} & 0 & 0 & \frac{25}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{175}{8}\end{matrix}\right]$$
,
and subtract it from other lines:
From 1 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}1 - - \frac{\sqrt{65}}{4} \frac{1}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} & 1 - 0 \frac{1}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} & - 0 \frac{1}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} & \frac{1}{2} - \frac{1}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) & - \frac{1}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right) - 1\end{matrix}\right] = \left[\begin{matrix}\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1 & 1 & 0 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - 1\end{matrix}\right]$$
you get
$$\left[\begin{matrix}\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1 & 1 & 0 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - 1\\- \frac{\sqrt{65}}{4} & 0 & 0 & \frac{25}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{175}{8}\\0 & 0 & -2 & 1 & -2\\- \frac{\sqrt{65}}{4} & 0 & 0 & - \frac{15}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{87}{8}\end{matrix}\right]$$
From 3 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- \frac{\left(-1\right) \frac{1}{4} \sqrt{65}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} & - \frac{0}{\frac{25}{16} - \frac{\sqrt{65}}{16}} & -2 - \frac{0}{\frac{25}{16} - \frac{\sqrt{65}}{16}} & 1 - \frac{\frac{25}{16} - \frac{\sqrt{65}}{16}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} - 2\end{matrix}\right] = \left[\begin{matrix}\frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} & 0 & -2 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} - 2\end{matrix}\right]$$
you get
$$\left[\begin{matrix}\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1 & 1 & 0 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - 1\\- \frac{\sqrt{65}}{4} & 0 & 0 & \frac{25}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{175}{8}\\\frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} & 0 & -2 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} - 2\\- \frac{\sqrt{65}}{4} & 0 & 0 & - \frac{15}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{87}{8}\end{matrix}\right]$$
From 4 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- - \frac{\sqrt{65}}{4} \frac{- \frac{15}{16} - \frac{\sqrt{65}}{16}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} - \frac{\sqrt{65}}{4} & - 0 \frac{- \frac{15}{16} - \frac{\sqrt{65}}{16}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} & - 0 \frac{- \frac{15}{16} - \frac{\sqrt{65}}{16}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} & \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) - \frac{- \frac{15}{16} - \frac{\sqrt{65}}{16}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) & \left(\frac{\sqrt{65}}{8} + \frac{87}{8}\right) - \frac{- \frac{15}{16} - \frac{\sqrt{65}}{16}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)\end{matrix}\right] = \left[\begin{matrix}\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4} & 0 & 0 & 0 & \frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1 & 1 & 0 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - 1\\- \frac{\sqrt{65}}{4} & 0 & 0 & \frac{25}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{175}{8}\\\frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} & 0 & -2 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} - 2\\\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4} & 0 & 0 & 0 & \frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\end{matrix}\right]$$
In 1 -th column
$$\left[\begin{matrix}\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1\\- \frac{\sqrt{65}}{4}\\\frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)}\\\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\end{matrix}\right]$$
let’s convert all the elements, except
4 -th element into zero.
- To do this, let’s take 4 -th line
$$\left[\begin{matrix}\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4} & 0 & 0 & 0 & \frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\end{matrix}\right]$$
,
and subtract it from other lines:
From 1 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- \frac{\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1}{\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}} \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right) + \left(\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1\right) & 1 - 0 \frac{\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1}{\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}} & - 0 \frac{\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1}{\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}} & - 0 \frac{\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1}{\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}} & \left(- \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - 1\right) - \frac{\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1}{\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}} \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)\end{matrix}\right] = \left[\begin{matrix}0 & 1 & 0 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - 1 - \frac{\left(\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1\right) \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}0 & 1 & 0 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - 1 - \frac{\left(\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1\right) \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}}\\- \frac{\sqrt{65}}{4} & 0 & 0 & \frac{25}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{175}{8}\\\frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} & 0 & -2 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} - 2\\\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4} & 0 & 0 & 0 & \frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\end{matrix}\right]$$
From 2 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- \frac{\sqrt{65}}{4} - - \frac{\sqrt{65}}{4 \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right) & - 0 \left(- \frac{\sqrt{65}}{4 \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)}\right) & - 0 \left(- \frac{\sqrt{65}}{4 \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)}\right) & - 0 \left(- \frac{\sqrt{65}}{4 \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)}\right) + \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) & - - \frac{\sqrt{65}}{4 \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right) + \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)\end{matrix}\right] = \left[\begin{matrix}0 & 0 & 0 & \frac{25}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65} \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{4 \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} + \frac{\sqrt{65}}{8} + \frac{175}{8}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}0 & 1 & 0 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - 1 - \frac{\left(\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1\right) \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}}\\0 & 0 & 0 & \frac{25}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65} \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{4 \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} + \frac{\sqrt{65}}{8} + \frac{175}{8}\\\frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} & 0 & -2 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} - 2\\\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4} & 0 & 0 & 0 & \frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\end{matrix}\right]$$
From 3 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- \frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right) + \frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} & - 0 \frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} & -2 - 0 \frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} & - 0 \frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} & \left(- \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} - 2\right) - \frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)\end{matrix}\right] = \left[\begin{matrix}0 & 0 & -2 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} - 2 - \frac{\sqrt{65} \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}0 & 1 & 0 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - 1 - \frac{\left(\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1\right) \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}}\\0 & 0 & 0 & \frac{25}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65} \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{4 \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} + \frac{\sqrt{65}}{8} + \frac{175}{8}\\0 & 0 & -2 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} - 2 - \frac{\sqrt{65} \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)}\\\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4} & 0 & 0 & 0 & \frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\end{matrix}\right]$$
It is almost ready, all we have to do is to find variables, solving the elementary equations:
$$x_{2} + \frac{\left(\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1\right) \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}} + 1 + \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} = 0$$
$$x_{4} \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) - \frac{175}{8} - \frac{\sqrt{65}}{8} - \frac{\sqrt{65} \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{4 \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} = 0$$
$$- 2 x_{3} + \frac{\sqrt{65} \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} + 2 + \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} = 0$$
$$x_{1} \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right) + \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}} - \frac{87}{8} - \frac{\sqrt{65}}{8} = 0$$
We get the answer:
$$x_{2} = - \frac{8}{5} + \frac{12 \sqrt{65}}{13}$$
$$x_{4} = \frac{22}{5}$$
$$x_{3} = \frac{16}{5}$$
$$x_{1} = - \frac{12 \sqrt{65}}{13} - \frac{8}{5}$$
$$c + \left(a + b\right) = 0$$
$$d + \left(c + \left(- a \frac{1 + \sqrt{65}}{8} - b \frac{1 - \sqrt{65}}{8}\right)\right) = 23$$
$$d + \left(- c + \left(a + b\right)\right) = -2$$
$$- d + \left(- a \frac{1 + \sqrt{65}}{8} - b \frac{1 - \sqrt{65}}{8}\right) = 11$$
We give the system of equations to the canonical form
$$a + b + c = 0$$
$$- \frac{\sqrt{65} a}{8} - \frac{a}{8} - \frac{b}{8} + \frac{\sqrt{65} b}{8} + c + d - 23 = 0$$
$$a + b - c + d = -2$$
$$- \frac{\sqrt{65} a}{8} - \frac{a}{8} - \frac{b}{8} + \frac{\sqrt{65} b}{8} - d - 11 = 0$$
Rewrite the system of linear equations as the matrix form
$$\left[\begin{matrix}1 & 1 & 1 & 0 & 0\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 1 & 1 & 23\\1 & 1 & -1 & 1 & -2\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 0 & -1 & 11\end{matrix}\right]$$
In 1 -th column
$$\left[\begin{matrix}1\\- \frac{\sqrt{65}}{8} - \frac{1}{8}\\1\\- \frac{\sqrt{65}}{8} - \frac{1}{8}\end{matrix}\right]$$
let’s convert all the elements, except
1 -th element into zero.
- To do this, let’s take 1 -th line
$$\left[\begin{matrix}1 & 1 & 1 & 0 & 0\end{matrix}\right]$$
,
and subtract it from other lines:
From 2 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}\left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right) - \left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right) & \left(- \frac{1}{8} + \frac{\sqrt{65}}{8}\right) - \left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right) & 1 - \left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right) & 1 - 0 \left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right) & 23 - 0 \left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right)\end{matrix}\right] = \left[\begin{matrix}0 & \frac{\sqrt{65}}{4} & \frac{\sqrt{65}}{8} + \frac{9}{8} & 1 & 23\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 1 & 1 & 0 & 0\\0 & \frac{\sqrt{65}}{4} & \frac{\sqrt{65}}{8} + \frac{9}{8} & 1 & 23\\1 & 1 & -1 & 1 & -2\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 0 & -1 & 11\end{matrix}\right]$$
From 3 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}-1 + 1 & -1 + 1 & -1 - 1 & \left(-1\right) 0 + 1 & -2 + \left(-1\right) 0\end{matrix}\right] = \left[\begin{matrix}0 & 0 & -2 & 1 & -2\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 1 & 1 & 0 & 0\\0 & \frac{\sqrt{65}}{4} & \frac{\sqrt{65}}{8} + \frac{9}{8} & 1 & 23\\0 & 0 & -2 & 1 & -2\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 0 & -1 & 11\end{matrix}\right]$$
From 4 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}\left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right) - \left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right) & \left(- \frac{1}{8} + \frac{\sqrt{65}}{8}\right) - \left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right) & - (- \frac{\sqrt{65}}{8} - \frac{1}{8}) & -1 - 0 \left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right) & 11 - 0 \left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right)\end{matrix}\right] = \left[\begin{matrix}0 & \frac{\sqrt{65}}{4} & \frac{1}{8} + \frac{\sqrt{65}}{8} & -1 & 11\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 1 & 1 & 0 & 0\\0 & \frac{\sqrt{65}}{4} & \frac{\sqrt{65}}{8} + \frac{9}{8} & 1 & 23\\0 & 0 & -2 & 1 & -2\\0 & \frac{\sqrt{65}}{4} & \frac{1}{8} + \frac{\sqrt{65}}{8} & -1 & 11\end{matrix}\right]$$
In 2 -th column
$$\left[\begin{matrix}1\\\frac{\sqrt{65}}{4}\\0\\\frac{\sqrt{65}}{4}\end{matrix}\right]$$
let’s convert all the elements, except
1 -th element into zero.
- To do this, let’s take 1 -th line
$$\left[\begin{matrix}1 & 1 & 1 & 0 & 0\end{matrix}\right]$$
,
and subtract it from other lines:
From 2 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- \frac{\sqrt{65}}{4} & - \frac{\sqrt{65}}{4} + \frac{\sqrt{65}}{4} & - \frac{\sqrt{65}}{4} + \left(\frac{\sqrt{65}}{8} + \frac{9}{8}\right) & 1 - 0 \frac{\sqrt{65}}{4} & 23 - 0 \frac{\sqrt{65}}{4}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{65}}{4} & 0 & \frac{9}{8} - \frac{\sqrt{65}}{8} & 1 & 23\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 1 & 1 & 0 & 0\\- \frac{\sqrt{65}}{4} & 0 & \frac{9}{8} - \frac{\sqrt{65}}{8} & 1 & 23\\0 & 0 & -2 & 1 & -2\\0 & \frac{\sqrt{65}}{4} & \frac{1}{8} + \frac{\sqrt{65}}{8} & -1 & 11\end{matrix}\right]$$
From 4 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- \frac{\sqrt{65}}{4} & - \frac{\sqrt{65}}{4} + \frac{\sqrt{65}}{4} & - \frac{\sqrt{65}}{4} + \left(\frac{1}{8} + \frac{\sqrt{65}}{8}\right) & -1 - 0 \frac{\sqrt{65}}{4} & 11 - 0 \frac{\sqrt{65}}{4}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{65}}{4} & 0 & \frac{1}{8} - \frac{\sqrt{65}}{8} & -1 & 11\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 1 & 1 & 0 & 0\\- \frac{\sqrt{65}}{4} & 0 & \frac{9}{8} - \frac{\sqrt{65}}{8} & 1 & 23\\0 & 0 & -2 & 1 & -2\\- \frac{\sqrt{65}}{4} & 0 & \frac{1}{8} - \frac{\sqrt{65}}{8} & -1 & 11\end{matrix}\right]$$
In 3 -th column
$$\left[\begin{matrix}1\\\frac{9}{8} - \frac{\sqrt{65}}{8}\\-2\\\frac{1}{8} - \frac{\sqrt{65}}{8}\end{matrix}\right]$$
let’s convert all the elements, except
3 -th element into zero.
- To do this, let’s take 3 -th line
$$\left[\begin{matrix}0 & 0 & -2 & 1 & -2\end{matrix}\right]$$
,
and subtract it from other lines:
From 1 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}1 - \frac{\left(-1\right) 0}{2} & 1 - \frac{\left(-1\right) 0}{2} & 1 - - -1 & - \frac{-1}{2} & - \frac{\left(-1\right) \left(-1\right) 2}{2}\end{matrix}\right] = \left[\begin{matrix}1 & 1 & 0 & \frac{1}{2} & -1\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 1 & 0 & \frac{1}{2} & -1\\- \frac{\sqrt{65}}{4} & 0 & \frac{9}{8} - \frac{\sqrt{65}}{8} & 1 & 23\\0 & 0 & -2 & 1 & -2\\- \frac{\sqrt{65}}{4} & 0 & \frac{1}{8} - \frac{\sqrt{65}}{8} & -1 & 11\end{matrix}\right]$$
From 2 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- \frac{\sqrt{65}}{4} - 0 \left(- \frac{9}{16} + \frac{\sqrt{65}}{16}\right) & - 0 \left(- \frac{9}{16} + \frac{\sqrt{65}}{16}\right) & - \left(-1\right) 2 \left(- \frac{9}{16} + \frac{\sqrt{65}}{16}\right) + \left(\frac{9}{8} - \frac{\sqrt{65}}{8}\right) & 1 - \left(- \frac{9}{16} + \frac{\sqrt{65}}{16}\right) & 23 - - 2 \left(- \frac{9}{16} + \frac{\sqrt{65}}{16}\right)\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{65}}{4} & 0 & 0 & \frac{25}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{175}{8}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 1 & 0 & \frac{1}{2} & -1\\- \frac{\sqrt{65}}{4} & 0 & 0 & \frac{25}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{175}{8}\\0 & 0 & -2 & 1 & -2\\- \frac{\sqrt{65}}{4} & 0 & \frac{1}{8} - \frac{\sqrt{65}}{8} & -1 & 11\end{matrix}\right]$$
From 4 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- \frac{\sqrt{65}}{4} - 0 \left(- \frac{1}{16} + \frac{\sqrt{65}}{16}\right) & - 0 \left(- \frac{1}{16} + \frac{\sqrt{65}}{16}\right) & \left(\frac{1}{8} - \frac{\sqrt{65}}{8}\right) - - 2 \left(- \frac{1}{16} + \frac{\sqrt{65}}{16}\right) & -1 - \left(- \frac{1}{16} + \frac{\sqrt{65}}{16}\right) & 11 - - 2 \left(- \frac{1}{16} + \frac{\sqrt{65}}{16}\right)\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{65}}{4} & 0 & 0 & - \frac{15}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{87}{8}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}1 & 1 & 0 & \frac{1}{2} & -1\\- \frac{\sqrt{65}}{4} & 0 & 0 & \frac{25}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{175}{8}\\0 & 0 & -2 & 1 & -2\\- \frac{\sqrt{65}}{4} & 0 & 0 & - \frac{15}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{87}{8}\end{matrix}\right]$$
In 4 -th column
$$\left[\begin{matrix}\frac{1}{2}\\\frac{25}{16} - \frac{\sqrt{65}}{16}\\1\\- \frac{15}{16} - \frac{\sqrt{65}}{16}\end{matrix}\right]$$
let’s convert all the elements, except
2 -th element into zero.
- To do this, let’s take 2 -th line
$$\left[\begin{matrix}- \frac{\sqrt{65}}{4} & 0 & 0 & \frac{25}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{175}{8}\end{matrix}\right]$$
,
and subtract it from other lines:
From 1 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}1 - - \frac{\sqrt{65}}{4} \frac{1}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} & 1 - 0 \frac{1}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} & - 0 \frac{1}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} & \frac{1}{2} - \frac{1}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) & - \frac{1}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right) - 1\end{matrix}\right] = \left[\begin{matrix}\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1 & 1 & 0 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - 1\end{matrix}\right]$$
you get
$$\left[\begin{matrix}\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1 & 1 & 0 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - 1\\- \frac{\sqrt{65}}{4} & 0 & 0 & \frac{25}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{175}{8}\\0 & 0 & -2 & 1 & -2\\- \frac{\sqrt{65}}{4} & 0 & 0 & - \frac{15}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{87}{8}\end{matrix}\right]$$
From 3 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- \frac{\left(-1\right) \frac{1}{4} \sqrt{65}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} & - \frac{0}{\frac{25}{16} - \frac{\sqrt{65}}{16}} & -2 - \frac{0}{\frac{25}{16} - \frac{\sqrt{65}}{16}} & 1 - \frac{\frac{25}{16} - \frac{\sqrt{65}}{16}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} - 2\end{matrix}\right] = \left[\begin{matrix}\frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} & 0 & -2 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} - 2\end{matrix}\right]$$
you get
$$\left[\begin{matrix}\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1 & 1 & 0 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - 1\\- \frac{\sqrt{65}}{4} & 0 & 0 & \frac{25}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{175}{8}\\\frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} & 0 & -2 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} - 2\\- \frac{\sqrt{65}}{4} & 0 & 0 & - \frac{15}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{87}{8}\end{matrix}\right]$$
From 4 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- - \frac{\sqrt{65}}{4} \frac{- \frac{15}{16} - \frac{\sqrt{65}}{16}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} - \frac{\sqrt{65}}{4} & - 0 \frac{- \frac{15}{16} - \frac{\sqrt{65}}{16}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} & - 0 \frac{- \frac{15}{16} - \frac{\sqrt{65}}{16}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} & \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) - \frac{- \frac{15}{16} - \frac{\sqrt{65}}{16}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) & \left(\frac{\sqrt{65}}{8} + \frac{87}{8}\right) - \frac{- \frac{15}{16} - \frac{\sqrt{65}}{16}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)\end{matrix}\right] = \left[\begin{matrix}\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4} & 0 & 0 & 0 & \frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1 & 1 & 0 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - 1\\- \frac{\sqrt{65}}{4} & 0 & 0 & \frac{25}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{175}{8}\\\frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} & 0 & -2 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} - 2\\\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4} & 0 & 0 & 0 & \frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\end{matrix}\right]$$
In 1 -th column
$$\left[\begin{matrix}\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1\\- \frac{\sqrt{65}}{4}\\\frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)}\\\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\end{matrix}\right]$$
let’s convert all the elements, except
4 -th element into zero.
- To do this, let’s take 4 -th line
$$\left[\begin{matrix}\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4} & 0 & 0 & 0 & \frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\end{matrix}\right]$$
,
and subtract it from other lines:
From 1 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- \frac{\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1}{\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}} \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right) + \left(\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1\right) & 1 - 0 \frac{\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1}{\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}} & - 0 \frac{\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1}{\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}} & - 0 \frac{\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1}{\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}} & \left(- \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - 1\right) - \frac{\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1}{\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}} \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)\end{matrix}\right] = \left[\begin{matrix}0 & 1 & 0 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - 1 - \frac{\left(\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1\right) \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}0 & 1 & 0 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - 1 - \frac{\left(\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1\right) \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}}\\- \frac{\sqrt{65}}{4} & 0 & 0 & \frac{25}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65}}{8} + \frac{175}{8}\\\frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} & 0 & -2 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} - 2\\\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4} & 0 & 0 & 0 & \frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\end{matrix}\right]$$
From 2 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- \frac{\sqrt{65}}{4} - - \frac{\sqrt{65}}{4 \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right) & - 0 \left(- \frac{\sqrt{65}}{4 \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)}\right) & - 0 \left(- \frac{\sqrt{65}}{4 \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)}\right) & - 0 \left(- \frac{\sqrt{65}}{4 \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)}\right) + \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) & - - \frac{\sqrt{65}}{4 \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right) + \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)\end{matrix}\right] = \left[\begin{matrix}0 & 0 & 0 & \frac{25}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65} \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{4 \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} + \frac{\sqrt{65}}{8} + \frac{175}{8}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}0 & 1 & 0 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - 1 - \frac{\left(\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1\right) \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}}\\0 & 0 & 0 & \frac{25}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65} \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{4 \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} + \frac{\sqrt{65}}{8} + \frac{175}{8}\\\frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} & 0 & -2 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} - 2\\\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4} & 0 & 0 & 0 & \frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\end{matrix}\right]$$
From 3 -th line. Let’s subtract it from this line:
$$\left[\begin{matrix}- \frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right) + \frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} & - 0 \frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} & -2 - 0 \frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} & - 0 \frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} & \left(- \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} - 2\right) - \frac{\sqrt{65}}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)\end{matrix}\right] = \left[\begin{matrix}0 & 0 & -2 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} - 2 - \frac{\sqrt{65} \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)}\end{matrix}\right]$$
you get
$$\left[\begin{matrix}0 & 1 & 0 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - 1 - \frac{\left(\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1\right) \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}}\\0 & 0 & 0 & \frac{25}{16} - \frac{\sqrt{65}}{16} & \frac{\sqrt{65} \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{4 \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} + \frac{\sqrt{65}}{8} + \frac{175}{8}\\0 & 0 & -2 & 0 & - \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} - 2 - \frac{\sqrt{65} \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)}\\\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4} & 0 & 0 & 0 & \frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\end{matrix}\right]$$
It is almost ready, all we have to do is to find variables, solving the elementary equations:
$$x_{2} + \frac{\left(\frac{\sqrt{65}}{8 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} + 1\right) \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}} + 1 + \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{2 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} = 0$$
$$x_{4} \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) - \frac{175}{8} - \frac{\sqrt{65}}{8} - \frac{\sqrt{65} \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{4 \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} = 0$$
$$- 2 x_{3} + \frac{\sqrt{65} \left(\frac{\sqrt{65}}{8} + \frac{87}{8} - \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right)} + 2 + \frac{\frac{\sqrt{65}}{8} + \frac{175}{8}}{\frac{25}{16} - \frac{\sqrt{65}}{16}} = 0$$
$$x_{1} \left(\frac{\sqrt{65} \left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right)}{4 \left(\frac{25}{16} - \frac{\sqrt{65}}{16}\right)} - \frac{\sqrt{65}}{4}\right) + \frac{\left(- \frac{15}{16} - \frac{\sqrt{65}}{16}\right) \left(\frac{\sqrt{65}}{8} + \frac{175}{8}\right)}{\frac{25}{16} - \frac{\sqrt{65}}{16}} - \frac{87}{8} - \frac{\sqrt{65}}{8} = 0$$
We get the answer:
$$x_{2} = - \frac{8}{5} + \frac{12 \sqrt{65}}{13}$$
$$x_{4} = \frac{22}{5}$$
$$x_{3} = \frac{16}{5}$$
$$x_{1} = - \frac{12 \sqrt{65}}{13} - \frac{8}{5}$$
Cramer's rule
$$c + \left(a + b\right) = 0$$
$$d + \left(c + \left(- a \frac{1 + \sqrt{65}}{8} - b \frac{1 - \sqrt{65}}{8}\right)\right) = 23$$
$$d + \left(- c + \left(a + b\right)\right) = -2$$
$$- d + \left(- a \frac{1 + \sqrt{65}}{8} - b \frac{1 - \sqrt{65}}{8}\right) = 11$$
We give the system of equations to the canonical form
$$a + b + c = 0$$
$$- \frac{\sqrt{65} a}{8} - \frac{a}{8} - \frac{b}{8} + \frac{\sqrt{65} b}{8} + c + d - 23 = 0$$
$$a + b - c + d = -2$$
$$- \frac{\sqrt{65} a}{8} - \frac{a}{8} - \frac{b}{8} + \frac{\sqrt{65} b}{8} - d - 11 = 0$$
Rewrite the system of linear equations as the matrix form
$$\left[\begin{matrix}x_{1} + x_{2} + x_{3} + 0 x_{4}\\x_{1} \left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right) + x_{2} \left(- \frac{1}{8} + \frac{\sqrt{65}}{8}\right) + x_{3} + x_{4}\\x_{1} + x_{2} - x_{3} + x_{4}\\x_{1} \left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right) + x_{2} \left(- \frac{1}{8} + \frac{\sqrt{65}}{8}\right) + 0 x_{3} - x_{4}\end{matrix}\right] = \left[\begin{matrix}0\\23\\-2\\11\end{matrix}\right]$$
- this is the system of equations that has the form
A*x = B
Let´s find a solution of this matrix equations using Cramer´s rule:
Since the determinant of the matrix:
$$A = \operatorname{det}{\left(\left[\begin{matrix}1 & 1 & 1 & 0\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 1 & 1\\1 & 1 & -1 & 1\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 0 & -1\end{matrix}\right] \right)} = \frac{5 \sqrt{65}}{4}$$
, then
The root xi is obtained by dividing the determinant of the matrix Ai. by the determinant of the matrix A.
( Ai we get it by replacement in the matrix A i-th column with column B )
$$x_{1} = \frac{4 \sqrt{65} \operatorname{det}{\left(\left[\begin{matrix}0 & 1 & 1 & 0\\23 & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 1 & 1\\-2 & 1 & -1 & 1\\11 & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 0 & -1\end{matrix}\right] \right)}}{325} = - \frac{4 \sqrt{65} \left(15 - \frac{2 \sqrt{65}}{5}\right)}{65} - \frac{16}{5}$$
=
$$- \frac{12 \sqrt{65}}{13} - \frac{8}{5}$$
$$x_{2} = \frac{4 \sqrt{65} \operatorname{det}{\left(\left[\begin{matrix}1 & 0 & 1 & 0\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & 23 & 1 & 1\\1 & -2 & -1 & 1\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & 11 & 0 & -1\end{matrix}\right] \right)}}{325} = \frac{4 \sqrt{65} \left(15 - \frac{2 \sqrt{65}}{5}\right)}{65}$$
=
$$- \frac{8}{5} + \frac{12 \sqrt{65}}{13}$$
$$x_{3} = \frac{4 \sqrt{65} \operatorname{det}{\left(\left[\begin{matrix}1 & 1 & 0 & 0\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 23 & 1\\1 & 1 & -2 & 1\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 11 & -1\end{matrix}\right] \right)}}{325} = \frac{16}{5}$$
$$x_{4} = \frac{4 \sqrt{65} \operatorname{det}{\left(\left[\begin{matrix}1 & 1 & 1 & 0\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 1 & 23\\1 & 1 & -1 & -2\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 0 & 11\end{matrix}\right] \right)}}{325} = \frac{22}{5}$$
$$d + \left(c + \left(- a \frac{1 + \sqrt{65}}{8} - b \frac{1 - \sqrt{65}}{8}\right)\right) = 23$$
$$d + \left(- c + \left(a + b\right)\right) = -2$$
$$- d + \left(- a \frac{1 + \sqrt{65}}{8} - b \frac{1 - \sqrt{65}}{8}\right) = 11$$
We give the system of equations to the canonical form
$$a + b + c = 0$$
$$- \frac{\sqrt{65} a}{8} - \frac{a}{8} - \frac{b}{8} + \frac{\sqrt{65} b}{8} + c + d - 23 = 0$$
$$a + b - c + d = -2$$
$$- \frac{\sqrt{65} a}{8} - \frac{a}{8} - \frac{b}{8} + \frac{\sqrt{65} b}{8} - d - 11 = 0$$
Rewrite the system of linear equations as the matrix form
$$\left[\begin{matrix}x_{1} + x_{2} + x_{3} + 0 x_{4}\\x_{1} \left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right) + x_{2} \left(- \frac{1}{8} + \frac{\sqrt{65}}{8}\right) + x_{3} + x_{4}\\x_{1} + x_{2} - x_{3} + x_{4}\\x_{1} \left(- \frac{\sqrt{65}}{8} - \frac{1}{8}\right) + x_{2} \left(- \frac{1}{8} + \frac{\sqrt{65}}{8}\right) + 0 x_{3} - x_{4}\end{matrix}\right] = \left[\begin{matrix}0\\23\\-2\\11\end{matrix}\right]$$
- this is the system of equations that has the form
A*x = B
Let´s find a solution of this matrix equations using Cramer´s rule:
Since the determinant of the matrix:
$$A = \operatorname{det}{\left(\left[\begin{matrix}1 & 1 & 1 & 0\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 1 & 1\\1 & 1 & -1 & 1\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 0 & -1\end{matrix}\right] \right)} = \frac{5 \sqrt{65}}{4}$$
, then
The root xi is obtained by dividing the determinant of the matrix Ai. by the determinant of the matrix A.
( Ai we get it by replacement in the matrix A i-th column with column B )
$$x_{1} = \frac{4 \sqrt{65} \operatorname{det}{\left(\left[\begin{matrix}0 & 1 & 1 & 0\\23 & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 1 & 1\\-2 & 1 & -1 & 1\\11 & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 0 & -1\end{matrix}\right] \right)}}{325} = - \frac{4 \sqrt{65} \left(15 - \frac{2 \sqrt{65}}{5}\right)}{65} - \frac{16}{5}$$
=
$$- \frac{12 \sqrt{65}}{13} - \frac{8}{5}$$
$$x_{2} = \frac{4 \sqrt{65} \operatorname{det}{\left(\left[\begin{matrix}1 & 0 & 1 & 0\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & 23 & 1 & 1\\1 & -2 & -1 & 1\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & 11 & 0 & -1\end{matrix}\right] \right)}}{325} = \frac{4 \sqrt{65} \left(15 - \frac{2 \sqrt{65}}{5}\right)}{65}$$
=
$$- \frac{8}{5} + \frac{12 \sqrt{65}}{13}$$
$$x_{3} = \frac{4 \sqrt{65} \operatorname{det}{\left(\left[\begin{matrix}1 & 1 & 0 & 0\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 23 & 1\\1 & 1 & -2 & 1\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 11 & -1\end{matrix}\right] \right)}}{325} = \frac{16}{5}$$
$$x_{4} = \frac{4 \sqrt{65} \operatorname{det}{\left(\left[\begin{matrix}1 & 1 & 1 & 0\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 1 & 23\\1 & 1 & -1 & -2\\- \frac{\sqrt{65}}{8} - \frac{1}{8} & - \frac{1}{8} + \frac{\sqrt{65}}{8} & 0 & 11\end{matrix}\right] \right)}}{325} = \frac{22}{5}$$
Numerical answer
[src]
a1 = -9.042084075352507 b1 = 5.842084075352507 c1 = 3.2 d1 = 4.4
a1 = -9.042084075352507 b1 = 5.842084075352507 c1 = 3.2 d1 = 4.4