## Task $2 / 69$

Give all values of $x$ and $y$ that satisfy the system of equations

$$
[x]+2 y=\frac{9}{2}, \quad[2 x]+3 y=\frac{31}{4}
$$

with $x \geq 0$. Here, $[z]$ denotes the greatest integer not greater than $z$.
Let $[x]=n$. It is necessary to distinguish two cases:

1.) $n \leq x < n+0.5$

Then $[2x] = [2n]$ and we obtain the system of equations

$$
n + 2y = 4.5 \quad, \quad 2n + 3y = 7.75
$$

with the solution $n=2, y=1.25$. For $x$, this yields the inequality $2 \leq x < 2.5$.

2.) $n+0.5 \leq x < n+1$

Then $[2x] = 2n+1$ and we obtain the system of equations

$$
n + 2y = 4.5 \quad, \quad 2n + 1 + 3y = 7.75
$$

with the solution $n=0, y=2.25$. For $x$, this yields the inequality $0.5 \leq x < 1$.

Therefore, the values $2 \leq x < 2.5 ; y=1.25$ and $0.5 \leq x < 1 ; y=2.25$ satisfy the given system of equations.