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

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.