11. $$ \left\{\begin{array}{l}y^{2}=x^{2}-9 \ 2 y=x-3\end{array} ight. $$

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11. $$ \left\{\begin{array}{l}y^{2}=x^{2}-9 \ 2 y=x-3\end{array}ight. $$

UpStudy AI · Accepted Answer

Solve the system of equations $$ y^{2}=x^{2}-9;2y=x-3 $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}y^{2}=x^{2}-9\2y=x-3\end{array}ight.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}y^{2}=x^{2}-9\x=2y+3\end{array}ight.$$
- step2: Substitute the value of $$x:$$
  $$y^{2}=\left(2y+3ight)^{2}-9$$
- step3: Simplify:
  $$y^{2}=4y^{2}+12y$$
- step4: Move the expression to the left side:
  $$y^{2}-\left(4y^{2}+12yight)=0$$
- step5: Subtract the terms:
  $$-3y^{2}-12y=0$$
- step6: Factor the expression:
  $$-3y\left(y+4ight)=0$$
- step7: Separate into possible cases:
  $$\begin{align}&-3y=0\&y+4=0\end{align}$$
- step8: Solve the equation:
  $$\begin{align}&y=0\&y=-4\end{align}$$
- step9: Calculate:
  $$y=0\cup y=-4$$
- step10: Rearrange the terms:
  $$\left\{ \begin{array}{l}x=2y+3\y=0\end{array}ight.\cup \left\{ \begin{array}{l}x=2y+3\y=-4\end{array}ight.$$
- step11: Calculate:
  $$\left\{ \begin{array}{l}x=3\y=0\end{array}ight.\cup \left\{ \begin{array}{l}x=-5\y=-4\end{array}ight.$$
- step12: Calculate:
  $$\left\{ \begin{array}{l}x=-5\y=-4\end{array}ight.\cup \left\{ \begin{array}{l}x=3\y=0\end{array}ight.$$
- step13: Check the solution:
  $$\left\{ \begin{array}{l}x=-5\y=-4\end{array}ight.\cup \left\{ \begin{array}{l}x=3\y=0\end{array}ight.$$
- step14: Rewrite:
  $$\left(x,yight) = \left(-5,-4ight)\cup \left(x,yight) = \left(3,0ight)$$
The solution to the system of equations is $$ (x,y) = (-5,-4) $$ or $$ (x,y) = (3,0) $$.
The solutions are $$ (x,y) = (-5,-4) $$ or $$ (x,y) = (3,0) $$.

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