### 11.32 Solve the system of equations

$$
\left\{\begin{array}{l}
2 x-3|y|=1 \\
|x|+2 y=4
\end{array}\right.
$$

Graph the functions (11.33-11.35):
11.32 Depending on various combinations of the signs of \(x\) and \(y\), the given system splits into four systems:

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-357.jpg?height=248&width=714&top_left_y=734&top_left_x=162)

\[
-7 y = -7 ; y = 1 > 0, x = 4 - 2 y = 2 > 0
\]

solution of the system \((2 ; 1)\);
2) \(\left\{\begin{array}{l}x0 - \text{no solutions;}\end{array}\right.\)

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-357.jpg?height=250&width=729&top_left_y=1590&top_left_x=162)
4) \(\left\{\begin{array}{l}x0, \\ 2 x - 3 y = 1, \\ -x + 2 y = 4|2| ;\end{array} \quad\left\{\begin{array}{l}x0, \\ 2 x - 3 y = 1, \\ -2 x + 4 y = 8 ;\end{array}\right.\right.\)

\(y = 9 ; x = 2 y - 4 = 14 > 0\) - no solutions.

Answer: \((2 ; 1)\).

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-358.jpg?height=370&width=344&top_left_y=536&top_left_x=255)
a)

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-358.jpg?height=358&width=421&top_left_y=527&top_left_x=659)

b)

Fig. 11.9