3. $$ \left\{\begin{array}{l}y-6=-3 x^{2}-5 \ y=2 x^{2}+1\end{array} ight. $$

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3. $$ \left\{\begin{array}{l}y-6=-3 x^{2}-5 \ y=2 x^{2}+1\end{array}ight. $$

UpStudy AI · Accepted Answer

Solve the system of equations $$ y-6=-3x^{2}-5;y=2x^{2}+1 $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}y-6=-3x^{2}-5\y=2x^{2}+1\end{array}ight.$$
- step1: Substitute the value of $$y:$$
  $$2x^{2}+1-6=-3x^{2}-5$$
- step2: Simplify:
  $$2x^{2}-5=-3x^{2}-5$$
- step3: Cancel equal terms:
  $$2x^{2}=-3x^{2}$$
- step4: Set both side equal to 0:
  $$\left\{ \begin{array}{l}2x^{2}=0\-3x^{2}=0\end{array}ight.$$
- step5: Solve the equation:
  $$\left\{ \begin{array}{l}x=0\x=0\end{array}ight.$$
- step6: Find the intersection:
  $$x=0$$
- step7: Substitute the value of $$x:$$
  $$y=2	imes 0^{2}+1$$
- step8: Substitute back:
  $$y=1$$
- step9: Calculate:
  $$\left\{ \begin{array}{l}x=0\y=1\end{array}ight.$$
- step10: Check the solution:
  $$\left\{ \begin{array}{l}x=0\y=1\end{array}ight.$$
- step11: Rewrite:
  $$\left(x,yight) = \left(0,1ight)$$
The solution to the system of equations is $$ (x, y) = (0, 1) $$.
The solution is $$ (x, y) = (0, 1) $$.

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