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

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

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

Perform the Gauss-Jordan elimination (for steps, see Gauss-Jordan elimination calculator): $$$\left[\begin{array}{cccc|c}1 & 0 & 1 & 1 & 1\\0 & 1 & - \sqrt{2} - 1 & -1 + \sqrt{2} & -1\\0 & 0 & 2 \sqrt{2} + 3 & 3 - 2 \sqrt{2} & 2\\0 & 0 & 0 & -8 + 6 \sqrt{2} & 2 \left(-2 + \sqrt{2}\right)\end{array}\right]$$$.

Back-substitute:

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

$$$c = \frac{2 - \left(3 - 2 \sqrt{2}\right) \left(- \sqrt{2} - 1\right)}{2 \sqrt{2} + 3} = -1 + \sqrt{2}$$$

$$$b = -1 - \left(-1 + \sqrt{2}\right) \left(- \sqrt{2} - 1\right) - \left(-1 + \sqrt{2}\right) \left(- \sqrt{2} - 1\right) = -1 + \left(-1 + \sqrt{2}\right) \left(1 + \sqrt{2}\right) + \left(1 - \sqrt{2}\right) \left(- \sqrt{2} - 1\right)$$$

$$$a = 1 - \left(1\right) \left(- \sqrt{2} - 1\right) - \left(1\right) \left(-1 + \sqrt{2}\right) - \left(0\right) \left(-1 + \left(-1 + \sqrt{2}\right) \left(1 + \sqrt{2}\right) + \left(1 - \sqrt{2}\right) \left(- \sqrt{2} - 1\right)\right) = 3$$$

$$$a = 3$$$A

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

$$$c = -1 + \sqrt{2}\approx 0.414213562373095$$$A

$$$d = - \sqrt{2} - 1\approx -2.414213562373095$$$A