Find all solutions of the system of equations. {x−2y=2y2−x2=2x+4

Step 1
To find:
The solution of the system of equation.
Given:
The system of equation $x-2y=2$ and ${\displaystyle y^2-x^2=2x+4}$.
Calculation:
Simplify the equation $x-2y=2$ as follows:
$x-2y=2$
$x=2+2y$
Substitute $x=2+2y$ in ${\displaystyle y^2-x^2=2x+4}$.
${\displaystyle y^2-{(2y+2)}^2=2(2y+2)+4}$
${\displaystyle y^2-4y^2-4-8y=4y+4+4}$
${\displaystyle -3y^2-4-8-8y-4y=0}$
${\displaystyle -3y^2-12-12y=0}$
Further simplify as follows:
${\displaystyle y^2+4y+4=0}$
${\displaystyle {(y+2)}^2=0}$
$y=-2$
Sbstitute $y=-2$ in $x=2+2y$.
$x=2+2(-2)$
$=2-4$
$=-2$
Step 2
Thus, the solution of the system of equation $x=-2$ and $y=-2.$