Answer
$(-1,1), (1,-1)$
Work Step by Step
We are given the system:
$\begin{cases}
x+y=0\\
x^2+y^2=2
\end{cases}$
We will use the substitution method. Solve Equation 1 for $y$ and substitute the expression of $y$ in Equation 2 to eliminate $y$ and determine $x$:
$\begin{cases}
y=-x\\
x^2+(-x)^2=2
\end{cases}$
$x^2+x^2=2$
$2x^2=2$
$x^2=1$
$x=\pm 1$
$x_1=-1$
$x_2=1$
Substitute each of the values of $x$ in the expression of $y$ to determine $y$:
$y=-x$
$x_1=-1\Rightarrow y_1=-(-1)=1$
$x_2=1\Rightarrow y_2=-1$
The system's solutions are:
$(-1,1), (1,-1)$
