# Find all solutions of the system of equations.\begin{cases}x-2y=2\\y^{2}

Brittney Lord 2021-09-29 Answered

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

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