## Task 4 - 191214

a) Show that the system of equations

$$
\begin{aligned}
7 x+100 y & =0 \\
0.069 x+y & =0.3
\end{aligned}
$$

has a unique solution \(\left(x_{0}, y_{0}\right)\), and determine this solution!

In the following, in equation (2) of the system (1), (2), the coefficient of \(x\) is changed within a given \(\delta\)-neighborhood of 0.069, i.e., for a given real \(\delta > 0\), a real number \(h\) is restricted to the interval

$$
-\delta \leq h \leq \delta
$$

and for each such \(h\), the system of equations

$$
\begin{aligned}
7 x+100 y & =0 \\
(0.069+h) x+y & =0.3
\end{aligned}
$$

is considered. The goal is to achieve that \(x_{0}\) changes by at most \(1\%\) due to this change in the coefficient 0.069. This is the meaning of the following tasks b), c).

First, the following is defined:

If for some \(h\) the system of equations (1), (4) has a unique solution, this solution is denoted by \(\left(x_{h} ; y_{h}\right)\). If this is the case for all \(h\) mentioned in (3) (for a given \(\delta > 0\)) and if there is among these values of \(h\) one for which the number

$$
\eta=\frac{\left|x_{0}-x_{h}\right|}{\left|x_{0}\right|}
$$

is as large as possible, then this largest value of \(\eta\) is denoted by \(\eta_{\max }\) (maximum relative error of \(x\) with respect to (3)).

b) Determine all \(\delta > 0\) for which a maximum relative error \(\eta_{\max }\) exists with respect to (3)!

c) Determine among the values of \(\delta\) found in b) all those for which \(\eta_{\max } \leq 0.01\) even holds!
Assume $(x ; y)$ is a solution (1), (4). Then it follows

$$
\begin{aligned}
7 x+100 y & =0 \\
(6.9+100 h) x+100 y & =30 \\
(0.1-100 h) x & =-30
\end{aligned}
$$

If $h=0.001$, this implies a contradiction.

If $h \neq 0.001$, then $x=-\frac{30}{0.1-100 h}$; from this and (1) we get

$$
y=-\frac{7}{100} x=\frac{2.1}{0.1-100 h}
$$

Therefore, the system (1), (4) can only have a solution if $h \neq 0.001$, and in that case, the solution is the given $x, y$. These indeed satisfy (1) and (4).

Thus, there is a unique solution to the system (1), (4) for all $h$ from (3) if and only if $\delta0)$, i.e.,

$$
\eta \leq \frac{100 \delta}{0.1-100 \delta}
$$

and for $h=\delta$, the equality holds. This proves the existence of the maximum relative error of $x$ with respect to (3) for every $\delta<0.001$, and it is

$$
\eta_{\max }=\frac{100 \delta}{0.1-100 \delta}
$$

On the other hand, if $\delta \geq 0.001$, a maximum relative error of $x$ with respect to (3) cannot exist because in (3) the value $h=0.001$ is also allowed, for which no solution $\left(x_{h} ; y_{h}\right)$ exists.

Therefore, the $\delta$ sought in b) are all those for which $0<\delta<0.001$ holds.

c) The requirement

$$
\frac{100 \delta}{0.1-100 \delta} \leq 0.01
$$

is equivalent to $100 \delta \leq 0.001-\delta$ for $0<\delta<0.001$, i.e., $\delta \leq \frac{0.001}{101}=\frac{1}{10100}$.

Therefore, the $\delta$ sought in c) are all those for which

$$
0<\delta \leq \frac{1}{10100} \quad(=0.000 \overline{0099})
$$

Taken from $[5]$

### 9.21.2 Second Round 1979, Class 12