We are given a system of two equations with two variables, $x$ and $y$: $x - y = -1$ $-1 = x^2 + 1$ We are asked to find the solutions for $x$ and $y$.

1. Problem Description

We are given a system of two equations with two variables, $x$ and $y$:

$x - y = -1$

$-1 = x^2 + 1$

We are asked to find the solutions for $x$ and $y$.

2. Solution Steps

First, let's solve the second equation for $x$.

$-1 = x^2 + 1$

Subtract 1 from both sides:

$-2 = x^2$

$x^2 = -2$

Since $x^2$ cannot be negative for real values of $x$, $x = \pm\sqrt{-2} = \pm i\sqrt{2}$.

So, $x = i\sqrt{2}$ or $x = -i\sqrt{2}$ where $i$ is the imaginary unit.

Now, let's solve the first equation for $y$:

$x - y = -1$

$x + 1 = y$

$y = x + 1$

If $x = i\sqrt{2}$, then $y = i\sqrt{2} + 1$.

If $x = -i\sqrt{2}$, then $y = -i\sqrt{2} + 1$.

Therefore, the solutions are $(x, y) = (i\sqrt{2}, 1+i\sqrt{2})$ and $(x, y) = (-i\sqrt{2}, 1-i\sqrt{2})$.

3. Final Answer

The solutions are $(x, y) = (i\sqrt{2}, 1+i\sqrt{2})$ and $(x, y) = (-i\sqrt{2}, 1-i\sqrt{2})$.