## Task $3 / 63$

In the system of linear equations

$$
0.9 x - 3.2 y + 10.1 = 0 \quad, \quad 1.1 x - 1.0 y + 0.7 = 0
$$

deviations of $\pm 0.05$ are permissible for the coefficients of the unknowns and for the absolute terms. Determine the maximum possible deviations, both upwards and downwards, for the solutions $x=3$ and $y=4$!
A system of equations of the form

$$
a x-b y+c=0 \quad, \quad d x-e y+f=0
$$

has the solutions

$$
x=\frac{e c-b f}{b d-a e} \quad, \quad y=\frac{c d-a f}{b d-a e}
$$

The signs are chosen such that the numerators and denominators are both positive.

The values of $x$ and $y$ are maximized when the numerators are maximized and the denominators are minimized; they are minimized when the numerators are minimized and the denominators are maximized. Therefore, $x$ is maximized when $a, e$, and $c$ are maximized, and $b, d$, and $f$ are minimized:

$$
x_{\max }=\frac{1.05 \cdot 10.15-3.15 \cdot 0.65}{3.15 \cdot 1.05-0.95 \cdot 1.05}=3.73
$$

and minimized when $a, e$, and $c$ are minimized, and $b, d$, and $f$ are maximized:

$$
x_{\min }=\frac{0.95 \cdot 10.05-3.25 \cdot 0.75}{3.25 \cdot 1.15-0.85 \cdot 0.95}=2.43
$$

For $y$, additional consideration is necessary, as the numerator and denominator conditions cannot be satisfied simultaneously: If $c$ and $e$ are maximized, and $b$ and $f$ are minimized, $y$ is maximized. The values of $d$ and $a$ affect the numerator and denominator in the same direction; $d$ affects the denominator more strongly than the numerator, so it is chosen to be minimal; $a$ also affects the denominator more strongly than the numerator, so it is chosen to be maximal:

$$
y_{\max }=\frac{10.15 \cdot 1.05-0.95 \cdot 0.65}{3.15 \cdot 1.05-0.95 \cdot 1.05}=4.35
$$

Similarly, it can be concluded that $y$ is minimized when $a, e$, and $c$ are minimized, and $b, d$, and $f$ are maximized.

$$
y_{\min }=\frac{10.15 \cdot 1.15-0.85 \cdot 0.75}{3.25 \cdot 1.15-0.85 \cdot 0.95}=3.73
$$

Result: $2.43 \leq x \leq 3.73$, maximum deviations $-0.57$ and $+0.73$ $3.73 \leq y \leq 4.35$, maximum deviations $-0.27$ and $+0.35$