We are given a system of two equations with two variables, $x$ and $y$: $x + y = 1$ $y = x^2 + 1$ We need to find the values of $x$ and $y$ that satisfy both equations.

1. Problem Description

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

$x + y = 1$

$y = x^2 + 1$

We need to find the values of $x$ and $y$ that satisfy both equations.

2. Solution Steps

We can substitute the second equation into the first equation to eliminate $y$.

$x + (x^2 + 1) = 1$

$x + x^2 + 1 = 1$

$x^2 + x = 0$

Factor out $x$:

$x(x + 1) = 0$

This gives us two possible solutions for $x$:

$x = 0$ or $x + 1 = 0$, which means $x = -1$.

Now we can find the corresponding values of $y$ for each value of $x$ using the equation $y = x^2 + 1$.

If $x = 0$, then $y = (0)^2 + 1 = 0 + 1 = 1$. So, one solution is $(x, y) = (0, 1)$.

If $x = -1$, then $y = (-1)^2 + 1 = 1 + 1 = 2$. So, the other solution is $(x, y) = (-1, 2)$.

3. Final Answer

The solutions are $(0, 1)$ and $(-1, 2)$.