Solve $$$\begin{cases} a + c + d + e = 0 \\ 2 a + b - \sqrt{2} d + \sqrt{2} e = 0 \\ - 2 a + 3 b - 4 c + d \left(-2 + \sqrt{2}\right) + e \left(-2 - \sqrt{2}\right) = 0 \\ - 2 a + b + 4 c + \sqrt{2} d - \sqrt{2} e = -1 \\ a - b - c + d \left(1 - \sqrt{2}\right) + e \left(1 + \sqrt{2}\right) = 0 \end{cases}$$$ for $$$a$$$, $$$b$$$, $$$c$$$, $$$d$$$, $$$e$$$

Solve $$$\begin{cases} a + c + d + e = 0 \\ 2 a + b - \sqrt{2} d + \sqrt{2} e = 0 \\ - 2 a + 3 b - 4 c + d \left(-2 + \sqrt{2}\right) + e \left(-2 - \sqrt{2}\right) = 0 \\ - 2 a + b + 4 c + \sqrt{2} d - \sqrt{2} e = -1 \\ a - b - c + d \left(1 - \sqrt{2}\right) + e \left(1 + \sqrt{2}\right) = 0 \end{cases}$$$ for $$$a$$$, $$$b$$$, $$$c$$$, $$$d$$$, $$$e$$$ using the Gauss-Jordan Elimination method.

Write down the augmented matrix: $$$\left[\begin{array}{ccccc|c}1 & 0 & 1 & 1 & 1 & 0\\2 & 1 & 0 & - \sqrt{2} & \sqrt{2} & 0\\-2 & 3 & -4 & -2 + \sqrt{2} & -2 - \sqrt{2} & 0\\-2 & 1 & 4 & \sqrt{2} & - \sqrt{2} & -1\\1 & -1 & -1 & 1 - \sqrt{2} & 1 + \sqrt{2} & 0\end{array}\right]$$$.

Perform the Gauss-Jordan elimination (for steps, see Gauss-Jordan elimination calculator): $$$\left[\begin{array}{ccccc|c}1 & 0 & 1 & 1 & 1 & 0\\0 & 1 & -2 & -2 - \sqrt{2} & -2 + \sqrt{2} & 0\\0 & 0 & 4 & 4 \sqrt{2} + 6 & 6 - 4 \sqrt{2} & 0\\0 & 0 & 0 & - 6 \sqrt{2} - 8 & -8 + 6 \sqrt{2} & -1\\0 & 0 & 0 & 0 & 24 - 16 \sqrt{2} & 1 - \sqrt{2}\end{array}\right]$$$.

Back-substitute:

$$$e = \frac{1 - \sqrt{2}}{24 - 16 \sqrt{2}} = - \frac{1 + \sqrt{2}}{8}$$$

$$$d = \frac{-1 - \left(-8 + 6 \sqrt{2}\right) \left(- \frac{1 + \sqrt{2}}{8}\right)}{- 6 \sqrt{2} - 8} = \frac{-1 + \sqrt{2}}{8}$$$

$$$c = \frac{0 - \left(- \frac{1 + \sqrt{2}}{8}\right) \left(6 - 4 \sqrt{2}\right) - \left(\frac{-1 + \sqrt{2}}{8}\right) \left(4 \sqrt{2} + 6\right)}{4} = - \frac{1}{8}$$$

$$$b = 0 - \left(-2 + \sqrt{2}\right) \left(- \frac{1 + \sqrt{2}}{8}\right) - \left(-2 - \sqrt{2}\right) \left(\frac{-1 + \sqrt{2}}{8}\right) - \left(-2\right) \left(- \frac{1}{8}\right) = - \frac{1}{4} + \frac{\left(-2 + \sqrt{2}\right) \left(1 + \sqrt{2}\right)}{8} - \frac{\left(-2 - \sqrt{2}\right) \left(-1 + \sqrt{2}\right)}{8}$$$

$$$a = 0 - \left(1\right) \left(- \frac{1 + \sqrt{2}}{8}\right) - \left(1\right) \left(\frac{-1 + \sqrt{2}}{8}\right) - \left(- \frac{1}{8}\right) \left(1\right) - \left(0\right) \left(- \frac{1}{4} + \frac{\left(-2 + \sqrt{2}\right) \left(1 + \sqrt{2}\right)}{8} - \frac{\left(-2 - \sqrt{2}\right) \left(-1 + \sqrt{2}\right)}{8}\right) = - \frac{-1 + \sqrt{2}}{8} + \frac{1}{8} + \frac{1 + \sqrt{2}}{8}$$$

$$$a = - \frac{-1 + \sqrt{2}}{8} + \frac{1}{8} + \frac{1 + \sqrt{2}}{8} = 0.375$$$A

$$$b = - \frac{1}{4} + \frac{\left(-2 + \sqrt{2}\right) \left(1 + \sqrt{2}\right)}{8} - \frac{\left(-2 - \sqrt{2}\right) \left(-1 + \sqrt{2}\right)}{8} = -0.25$$$A

$$$c = - \frac{1}{8} = -0.125$$$A

$$$d = \frac{-1 + \sqrt{2}}{8}\approx 0.051776695296637$$$A

$$$e = - \frac{1 + \sqrt{2}}{8}\approx -0.301776695296637$$$A