Answer
$(-1,-1), \left(\dfrac{1}{2},2\right)$
Work Step by Step
We are given the system:
$\begin{cases}
y=2x+1\\
xy=1
\end{cases}$
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 $x$:
$\begin{cases}
y=2x+1\\
x(2x+1)=1
\end{cases}$
$2x^2+x=1$
$2x^2+x-1=0$
$2x^2+2x-x-1=0$
$2x(x+1)-(x+1)=0$
$(x+1)(2x-1)=0$
$x+1=0\Rightarrow x_1=-1$
$2x-1=0\Rightarrow x_2=\dfrac{1}{2}$
Substitute each of the values of $x$ in the expression of $y$ to determine $y$:
$y=2x+1$
$x_1=-1\Rightarrow y_1=2(-1)+1=-1$
$x_2=\dfrac{1}{2}\Rightarrow y_2=2\left(\dfrac{1}{2}\right)+1=2$
The system's solutions are:
$(-1,-1), \left(\dfrac{1}{2},2\right)$
