$$ 43 \left\{ \begin{array} { l } { y = x ^ { 2 } - 4 x + 7 } \ { y = - x ^ { 2 } - 6 x - 11 } \end{array} ight. $$

Question

$$ 43 \left\{ \begin{array} { l } { y = x ^ { 2 } - 4 x + 7 } \ { y = - x ^ { 2 } - 6 x - 11 } \end{array} ight. $$

UpStudy AI · Accepted Answer

Solve the system of equations $$ y=x^2-4x+7;-x^2-6x-11=y $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}y=x^{2}-4x+7\-x^{2}-6x-11=y\end{array}ight.$$
- step1: Substitute the value of $$y:$$
  $$-x^{2}-6x-11=x^{2}-4x+7$$
- step2: Move the expression to the left side:
  $$-x^{2}-6x-11-\left(x^{2}-4x+7ight)=0$$
- step3: Subtract the terms:
  $$-2x^{2}-2x-18=0$$
- step4: Multiply both sides:
  $$2x^{2}+2x+18=0$$
- step5: Solve using the quadratic formula:
  $$x=\frac{-2\pm \sqrt{2^{2}-4	imes 2	imes 18}}{2	imes 2}$$
- step6: Simplify the expression:
  $$x=\frac{-2\pm \sqrt{2^{2}-4	imes 2	imes 18}}{4}$$
- step7: Simplify the expression:
  $$x=\frac{-2\pm \sqrt{-140}}{4}$$
- step8: The expression is undefined:
  $$x 
otin \mathbb{R}$$
- step9: The system of equations has no solution in the set of real numbers:
  $$\left(x,yight) 
otin \mathbb{R}^{2}$$
- step10: Alternative Form:
  $$	extrm{No real solution}$$
The system of equations does not have a real solution.
The system of equations has no real solution.

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