Brittney Lord

2021-09-29

Find all solutions of the system of equations.
$\left\{\begin{array}{l}x-2y=2\\ {y}^{2}-{x}^{2}=2x+4\end{array}$

Gennenzip

Expert

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

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