Solve for $$ x $$ and $$ y $$ simultaneously: $$ x^{2}-y^{2}=25 ; x y8 $$

Question

Solve for $$ x $$ and $$ y $$ simultaneously: $$ x^{2}-y^{2}=25 ; x y>8 $$

UpStudy AI · Accepted Answer

The system of equations provided cannot be directly solved using the available methods. Let's try to solve it step by step.

Given equations:
1. $$ x^{2}-y^{2}=25 $$
2. $$ xy>8 $$

From equation 1, we can rewrite it as:
$$ x^{2}-y^{2}=25 $$
$$ (x+y)(x-y)=25 $$

Now, we have two cases to consider:
1. $$ x+y=5 $$ and $$ x-y=5 $$
2. $$ x+y=-5 $$ and $$ x-y=-5 $$

Let's solve these cases separately.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}x+y=5\x-y=5\end{array}\right.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}x=5-y\x-y=5\end{array}\right.$$
- step2: Substitute the value of $$x:$$
  $$5-y-y=5$$
- step3: Subtract the terms:
  $$5-2y=5$$
- step4: Move the constant to the right side:
  $$-2y=5-5$$
- step5: Subtract the terms:
  $$-2y=0$$
- step6: Change the signs:
  $$2y=0$$
- step7: Rewrite the expression:
  $$y=0$$
- step8: Substitute the value of $$y:$$
  $$x=5-0$$
- step9: Simplify:
  $$x=5$$
- step10: Calculate:
  $$\left\{ \begin{array}{l}x=5\y=0\end{array}\right.$$
- step11: Check the solution:
  $$\left\{ \begin{array}{l}x=5\y=0\end{array}\right.$$
- step12: Rewrite:
  $$\left(x,y\right) = \left(5,0\right)$$
Solve the system of equations $$ x+y=-5 ; x-y=-5 $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}x+y=-5\x-y=-5\end{array}\right.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}x=-5-y\x-y=-5\end{array}\right.$$
- step2: Substitute the value of $$x:$$
  $$-5-y-y=-5$$
- step3: Subtract the terms:
  $$-5-2y=-5$$
- step4: Move the constant to the right side:
  $$-2y=-5+5$$
- step5: Add the numbers:
  $$-2y=0$$
- step6: Change the signs:
  $$2y=0$$
- step7: Rewrite the expression:
  $$y=0$$
- step8: Substitute the value of $$y:$$
  $$x=-5-0$$
- step9: Simplify:
  $$x=-5$$
- step10: Calculate:
  $$\left\{ \begin{array}{l}x=-5\y=0\end{array}\right.$$
- step11: Check the solution:
  $$\left\{ \begin{array}{l}x=-5\y=0\end{array}\right.$$
- step12: Rewrite:
  $$\left(x,y\right) = \left(-5,0\right)$$
The solutions to the system of equations are:
1. $$ x=5 $$ and $$ y=0 $$
2. $$ x=-5 $$ and $$ y=0 $$

However, we also need to consider the condition $$ xy>8 $$. Let's check which of these solutions satisfy this condition.
Check the inequality by following steps:
- step0: Check the inequality is true or false:
  $$5\times 0>8$$
- step1: Multiply:
  $$0>8$$
- step2: Check the inequality:
  $$\textrm{false}$$
The solution $$ x=5 $$ and $$ y=0 $$ does not satisfy the condition $$ xy>8 $$.

Now, let's check the second solution $$ x=-5 $$ and $$ y=0 $$ for the condition $$ xy>8 $$.
Check the inequality by following steps:
- step0: Check the inequality is true or false:
  $$-5\times 0>8$$
- step1: Multiply:
  $$0>8$$
- step2: Check the inequality:
  $$\textrm{false}$$
The solution $$ x=-5 $$ and $$ y=0 $$ also does not satisfy the condition $$ xy>8 $$.

Therefore, there are no solutions that satisfy both equations $$ x^{2}-y^{2}=25 $$ and $$ xy>8 $$.
There are no solutions that satisfy both equations $$ x^{2}-y^{2}=25 $$ and $$ xy>8 $$.

Solve for and simultaneously:

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