Answer
$(1,3), (-3,-1)$
Work Step by Step
We are given the system:
$\begin{cases}
y=x+2\\
x^2+y^2=10
\end{cases}$
We will use the substitution method. Equation 1 is solved for $y$. Substitute the expression of $y$ in Equation 2 to eliminate $y$ and determine $x$:
$\begin{cases}
y=x+2\\
x^2+(x+2)^2=10
\end{cases}$
$x^2+x^2+4x+4=10$
$2x^2+4x+4-10=0$
$2x^2+4x-6=0$
$2(x^2+2x-3)=0$
$x^2+2x-3=0$
$x^2-x+3x-3=0$
$x(x-1)+3(x-1)=0$
$(x-1)(x+3)=0$
$x-1=0\Rightarrow x_1=1$
$x+3=0\Rightarrow x_2=-3$
Substitute each of the values of $x$ in the expression of $y$ to determine $y$:
$y=x+2$
$x_1=1\Rightarrow y_1=1+2=3$
$x_2=-3\Rightarrow y_2=-3+2=-1$
The system's solutions are:
$(1,3), (-3,-1)$
