## 54. System of Linear Equations.

- This system of $n$ linear equations with $n$ unknowns has an interesting property, - said the Great Mathematician.

- Oh my God! - exclaimed the Poor Thing. - What is this property?
- Pay attention, - said the Great Mathematician, - the coefficients form an arithmetic progression.
- After your explanations, everything becomes so clear, - said the Poor Thing. - You mean that this system is similar to $6 x+9 y=12$ and $15 x+18 y=21 ?$
- Exactly, - agreed the Great Mathematician, taking his bassoon out of the case. - By the way, this system has a unique solution. Can you find it?
- Oh my God! - exclaimed the Poor Thing. - I am completely puzzled.

And you?
54. If $n \leqslant 3$, then the equations of our system are dependent and, consequently, its solution is not unique*. Therefore, $n=1$. Consequently, $n=2$.

Since our system has the form

$$
\begin{gathered}
a x+(a+d) y=a+2 d \\
(a+3 d) x+(a+4 d) y=a+5 d
\end{gathered}
$$

then

$$
x+y=1 \quad \text { and } \quad x=-1, \quad y=2
$$

[D. Rothman, A. M. M., 70, 93 (January 1963).]

In fact, any system of two equations belonging to the family

$$
(a+3 k d) x+[a+(3 k+1) d] y=a+(3 k+2) d
$$

has the same unique solution.