$$ \left. \begin{array} { l } { 102 ^ { x } - 2 ^ { y + 2 } = 0 } \ { x ^ { 2 } + 2 x y + y ^ { 2 } = 36 } \end{array} ight. $$

Question

$$ \left. \begin{array} { l } { 102 ^ { x } - 2 ^ { y + 2 } = 0 } \ { x ^ { 2 } + 2 x y + y ^ { 2 } = 36 } \end{array} ight. $$

UpStudy AI · Accepted Answer

Solve the equation by following steps:
- step0: Solve for $$y$$:
  $$102^{x}-2^{y+2}=0$$
- step1: Move the expression to the right side:
  $$-2^{y+2}=0-102^{x}$$
- step2: Remove 0:
  $$-2^{y+2}=-102^{x}$$
- step3: Change the signs:
  $$2^{y+2}=102^{x}$$
- step4: Take the logarithm of both sides:
  $$\log_{2}{\left(2^{y+2}\right)}=\log_{2}{\left(102^{x}\right)}$$
- step5: Evaluate the logarithm:
  $$y+2=\log_{2}{\left(102^{x}\right)}$$
- step6: Simplify:
  $$y+2=x\log_{2}{\left(102\right)}$$
- step7: Move the constant to the right side:
  $$y=x\log_{2}{\left(102\right)}-2$$
- step8: Simplify:
  $$y=x+x\log_{2}{\left(51\right)}-2$$
Solve the equation $$ x^{2} + 2xy + y^{2} = 36 $$.
Solve the equation by following steps:
- step0: Solve for $$y$$:
  $$x^{2}+2xy+y^{2}=36$$
- step1: Move the expression to the left side:
  $$x^{2}+2xy+y^{2}-36=0$$
- step2: Simplify:
  $$x^{2}-36+2xy+y^{2}=0$$
- step3: Factor the expression:
  $$\left(x+6+y\right)\left(x-6+y\right)=0$$
- step4: Separate into possible cases:
  $$\begin{align}&x+6+y=0\&x-6+y=0\end{align}$$
- step5: Solve the equation:
  $$\begin{align}&y=-x-6\&y=-x+6\end{align}$$
Solve the system of equations $$ y=x+x\log_{2}{(51)}-2; y=-x-6; y=-x+6 $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}y=x+x\log_{2}{\left(51\right)}-2\y=-x-6\y=-x+6\end{array}\right.$$
- step1: Calculate:
  $$\left\{ \begin{array}{l}y=x+\log_{2}{\left(51\right)}\times x-2\y=-x-6\y=-x+6\end{array}\right.$$
- step2: Substitute the value of $$y:$$
  $$\left\{ \begin{array}{l}x+\log_{2}{\left(51\right)}\times x-2=-x-6\x+\log_{2}{\left(51\right)}\times x-2=-x+6\end{array}\right.$$
- step3: Solve the equation:
  $$\left\{ \begin{array}{l}x=-\frac{4}{2+\log_{2}{\left(51\right)}}\x+\log_{2}{\left(51\right)}\times x-2=-x+6\end{array}\right.$$
- step4: Substitute the value of $$x:$$
  $$-\frac{4}{2+\log_{2}{\left(51\right)}}+\log_{2}{\left(51\right)}\times \left(-\frac{4}{2+\log_{2}{\left(51\right)}}\right)-2=-\left(-\frac{4}{2+\log_{2}{\left(51\right)}}\right)+6$$
- step5: Simplify:
  $$-\frac{8+8\log_{204}{\left(51\right)}+4\log_{2}{\left(51\right)}\times \log_{204}{\left(51\right)}+2\log_{2}{\left(51\right)}}{2+\log_{2}{\left(51\right)}}=\log_{204}{\left(65536\times 51^{6}\right)}$$
- step6: Calculate:
  $$\left\{ \begin{array}{l}x \in \varnothing \y \in \varnothing \end{array}\right.$$
- step7: Rewrite:
  $$(x, y) \in \varnothing$$
Let's solve the system of equations step by step.

We have the following equations:

1. $$ 102^{x} - 2^{y + 2} = 0 $$
2. $$ x^{2} + 2xy + y^{2} = 36 $$

### Step 1: Solve the first equation for $$ y $$

From the first equation, we can express $$ y $$ in terms of $$ x $$:

$$
102^{x} = 2^{y + 2}
$$

Taking logarithm base 2 on both sides:

$$
\log_{2}(102^{x}) = y + 2
$$

This simplifies to:

$$
x \log_{2}(102) = y + 2
$$

Thus, we can express $$ y $$ as:

$$
y = x \log_{2}(102) - 2
$$

### Step 2: Substitute $$ y $$ into the second equation

Now, we substitute $$ y $$ into the second equation:

$$
x^{2} + 2x(x \log_{2}(102) - 2) + (x \log_{2}(102) - 2)^{2} = 36
$$

### Step 3: Solve the second equation

After substituting, we can simplify and solve for $$ x $$. However, I have already computed the results for $$ y $$ in terms of $$ x $$ and found two forms for $$ y $$:

1. $$ y = -x - 6 $$
2. $$ y = -x + 6 $$

### Step 4: Solve the system of equations

Now, we need to check if there are any solutions between the two forms of $$ y $$:

1. $$ y = x + x \log_{2}(51) - 2 $$
2. $$ y = -x - 6 $$
3. $$ y = -x + 6 $$

After checking these equations, we find that there are no solutions that satisfy both equations simultaneously.

### Conclusion

The final result is that there are no solutions to the system of equations:

$$
(x, y) \in \varnothing
$$

Thus, the system has no solution.
There are no solutions to the system of equations.

\( \left. \begin{array} { l } { 102 ^ { x } - 2 ^ { y + 2 } = 0 } \\ { x ^ { 2 } + 2 x y + y ^ { 2 } = 36 } \end{array} \right. \)

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