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

$$
x_{1}+7 x_{2}-8 x_{3}+9 x_{4}=4 \text {. }
$$
Solution. The ranks of the main matrix of the system and the extended matrix of the system are equal to one. Since the number of unknowns is four, the number of basic unknowns is one (let the basic unknown be $x_{1}$), and the number of free unknowns is three (let the free unknowns be $x_{2}, x_{3}, x_{4}$). Rewrite the system as follows:

$$
x_{1}=-7 x_{2}+8 x_{3}-9 x_{4}+4 .
$$

By equating the column of free terms

$$
\left(\begin{array}{l}
x_{2} \\
x_{3} \\
x_{4}
\end{array}\right)=\left(\begin{array}{l}
1 \\
0 \\
0
\end{array}\right), \quad X^{\text {(general hom. })}=\left(\begin{array}{l}
x_{2} \\
x_{3} \\
x_{4}
\end{array}\right)=\left(\begin{array}{l}
0 \\
1 \\
0
\end{array}\right),\left(\begin{array}{l}
x_{2} \\
x_{3} \\
x_{1}
\end{array}\right)=\left(\begin{array}{l}
0 \\
0 \\
1
\end{array}\right)
$$

we obtain the fundamental system of solutions of the homogeneous system $x_{1}+7 x_{2}-8 x_{3}+9 x_{4}=0$ :

$$
X^{(1)}=\left(\begin{array}{c}
-7 \\
1 \\
0 \\
0
\end{array}\right), X^{(2)}=\left(\begin{array}{l}
8 \\
0 \\
1 \\
0
\end{array}\right), \quad X^{(3)}=\left(\begin{array}{c}
-9 \\
0 \\
0 \\
1
\end{array}\right)
$$

The general solution of the corresponding homogeneous system:

$$
X^{\text {(general hom.) }}=\alpha_{1}\left(\begin{array}{c}
-7 \\
1 \\
0 \\
0
\end{array}\right)+\alpha_{2}\left(\begin{array}{l}
8 \\
0 \\
1 \\
0
\end{array}\right)+\alpha_{3}\left(\begin{array}{l}
9 \\
0 \\
0 \\
1
\end{array}\right)
$$

where $\alpha_{1}, \alpha_{2}, \alpha_{3}$ are any real numbers. Now let's find a particular solution of the original non-homogeneous system

$$
x_{1}+7 x_{2}-8 x_{3}+9 x_{4}=4 .
$$

For this, set all free unknowns to zero: $x_{1}+7 \cdot 0-8 \cdot 0+9 \cdot 0=4$. We get that the particular solution has the form

$$
X^{\text {(particular non-hom.) }}=\left(\begin{array}{l}
x_{1}=4 \\
x_{2}=0 \\
x_{3}=0 \\
x_{4}=0
\end{array}\right)
$$

The general solution of the non-homogeneous system is the sum of its particular solution and the general solution of the corresponding homogeneous system:

$$
\begin{aligned}
& X^{\text {(general non-hom.) }}=X^{\text {(general hom.) }}+X^{\text {(particular non-hom.) }}= \\
& =\alpha_{1}\left(\begin{array}{c}
-7 \\
1 \\
0 \\
0
\end{array}\right)+\alpha_{2}\left(\begin{array}{l}
8 \\
0 \\
1 \\
0
\end{array}\right)+\alpha_{3}\left(\begin{array}{l}
9 \\
0 \\
0 \\
1
\end{array}\right)+\left(\begin{array}{l}
4 \\
0 \\
0 \\
0
\end{array}\right)
\end{aligned}
$$