Example 1. Solve the system of equations

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

$$
\begin{gathered}
\Delta=\left|\begin{array}{rr}
2 & -1 \\
3 & 4
\end{array}\right|=2 \cdot 4-3 \cdot(-1)=8+3=11 \\
\Delta_{1}=\left|\begin{array}{rr}
3 & -1 \\
7 & 4
\end{array}\right|=12-(-7)=19, \quad \Delta_{2}=\left|\begin{array}{rr}
2 & -3 \\
3 & 7
\end{array}\right|=14-9=5
\end{gathered}
$$

By Cramer's formulas: $x_{1}=\frac{19}{11}, x_{2}=\frac{5}{11}$, or $X=\left(\frac{19}{11}, \frac{5}{11}\right)$.