1.25. Find the general solution of the non-homogeneous system

\[
\left\{\begin{array}{l}
x_{1}+7 x_{2}-8 x_{3}+9 x_{4}=1 \\
2 x_{1}-3 x_{2}+3 x_{3}-2 x_{4}=1 \\
4 x_{1}+11 x_{2}-13 x_{3}+16 x_{4}=3 \\
7 x_{1}-2 x_{2}+x_{3}+3 x_{4}=4
\end{array}\right.
\]
Solution. The main matrix of the system coincides with the one considered in Example 1.23. It is not difficult to show that the ranks of the augmented matrix of the system and the main matrix of the system are equal to 2, and the system is consistent. The system is equivalent to the following:

$$
\left\{\begin{array}{l}
x_{1}+7 x_{2}=1+8 x_{3}-9 x_{4} \\
2 x_{1}-3 x_{2}=1-3 x_{3}+2 x_{4}
\end{array} .\right.
$$

The general solution of the corresponding homogeneous system is obtained in Example 1.23, and it looks like this:

$$
X^{\text {(general homogeneous) }}=\alpha\left(\begin{array}{c}
\frac{3}{17} \\
\frac{19}{17} \\
1 \\
0
\end{array}\right)+\beta\left(\begin{array}{c}
-\frac{13}{17} \\
-\frac{20}{17} \\
0 \\
1
\end{array}\right)=\left(\begin{array}{c}
\alpha \cdot \frac{3}{17}+\beta \cdot\left(-\frac{13}{17}\right) \\
\alpha \cdot\left(\frac{19}{17}\right)+\beta \cdot\left(-\frac{20}{17}\right) \\
\alpha \cdot 1+\beta \cdot 0 \\
\alpha \cdot 0+\beta \cdot 1
\end{array}\right)
$$

To find a particular solution, we set the free variables equal to zero:

$$
\left\{\begin{array}{l}
x_{1}+7 x_{2}=1+8 \cdot 0-9 \cdot 0=1 \\
2 x_{1}-3 x_{2}=1-3 \cdot 0+2 \cdot 0=1
\end{array}\right.
$$

Using the inverse matrix for the main matrix of this system (it is found in Example 1.23), we get:

$$
X=\binom{x_{1}}{x_{2}}=A^{-1} \cdot B=\left(\begin{array}{cc}
\frac{3}{17} & \frac{7}{17} \\
\frac{2}{17} & -\frac{1}{17}
\end{array}\right) \cdot\binom{1}{1}=\binom{\frac{10}{17}}{\frac{1}{17}}
$$

from which it follows that

$$
X^{\text {(particular non-homogeneous) }}=\left(\begin{array}{c}
\frac{10}{17} \\
\frac{1}{17} \\
0 \\
0
\end{array}\right)
$$

Therefore,

$$
X^{\text {(general non-homogeneous) }}=\alpha\left(\begin{array}{c}
\frac{3}{17} \\
\frac{19}{17} \\
1 \\
0
\end{array}\right)+\beta\left(\begin{array}{c}
-\frac{13}{17} \\
-\frac{20}{17} \\
0 \\
1
\end{array}\right)+\left(\begin{array}{c}
\frac{10}{17} \\
\frac{1}{17} \\
0 \\
0
\end{array}\right)=\left(\begin{array}{c}
\alpha \cdot \frac{3}{17}+\beta \cdot\left(-\frac{13}{17}\right)+\frac{10}{17} \\
\alpha \cdot\left(\frac{19}{17}\right)+\beta \cdot\left(-\frac{20}{17}\right)+\frac{1}{17} \\
\alpha \cdot 1+\beta \cdot 0 \\
\alpha \cdot 0+\beta \cdot 1
\end{array}\right)
$$