4. $$ \left\{\begin{array}{l}y-14=8(x+2) \ y+11=(x+7)^{2}\end{array} ight. $$

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4. $$ \left\{\begin{array}{l}y-14=8(x+2) \ y+11=(x+7)^{2}\end{array}ight. $$

UpStudy AI · Accepted Answer

Let's simplify the expression $$ \left\{\begin{array}{l}y-14=8(x+2) \ y+11=(x+7)^{2}\end{array}ight. $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}y-14=8\left(x+2ight)\y+11=\left(x+7ight)^{2}\end{array}ight.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}y=8x+30\y+11=\left(x+7ight)^{2}\end{array}ight.$$
- step2: Substitute the value of $$y:$$
  $$8x+30+11=\left(x+7ight)^{2}$$
- step3: Simplify:
  $$8x+41=\left(x+7ight)^{2}$$
- step4: Move the expression to the left side:
  $$8x+41-\left(x+7ight)^{2}=0$$
- step5: Calculate:
  $$-6x-8-x^{2}=0$$
- step6: Factor the expression:
  $$-\left(2+xight)\left(4+xight)=0$$
- step7: Divide the terms:
  $$\left(2+xight)\left(4+xight)=0$$
- step8: Separate into possible cases:
  $$\begin{align}&2+x=0\&4+x=0\end{align}$$
- step9: Solve the equation:
  $$\begin{align}&x=-2\&x=-4\end{align}$$
- step10: Calculate:
  $$x=-2\cup x=-4$$
- step11: Rearrange the terms:
  $$\left\{ \begin{array}{l}x=-2\y=8x+30\end{array}ight.\cup \left\{ \begin{array}{l}x=-4\y=8x+30\end{array}ight.$$
- step12: Calculate:
  $$\left\{ \begin{array}{l}x=-2\y=14\end{array}ight.\cup \left\{ \begin{array}{l}x=-4\y=-2\end{array}ight.$$
- step13: Check the solution:
  $$\left\{ \begin{array}{l}x=-2\y=14\end{array}ight.\cup \left\{ \begin{array}{l}x=-4\y=-2\end{array}ight.$$
- step14: Rewrite:
  $$\left(x,yight) = \left(-2,14ight)\cup \left(x,yight) = \left(-4,-2ight)$$
The solutions for the system of equations $$ \left\{\begin{array}{l}y-14=8(x+2) \ y+11=(x+7)^{2}\end{array}ight. $$ are $$ (x,y) = (-2,14) $$ and $$ (x,y) = (-4,-2) $$.

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