We are given the following system of equations:

$2^{x+1} \cdot 3^y = 144$ $3^{x+2} \cdot 2^y = 27$

Step 1: Simplify both equations

1. First equation: $2^{x+1} \cdot 3^y = 144$ Factor 144: $144 = 2^4 \cdot 3^2$ So the first equation becomes: $2^{x+1} \cdot 3^y = 2^4 \cdot 3^2$ Equating powers of 2 and powers of 3: $x+1 = 4 \quad \text{and} \quad y = 2$ Solving for $x$: $x = 3$

2. Second equation: $3^{x+2} \cdot 2^y = 27$ Factor 27: $27 = 3^3$ So the second equation becomes: $3^{x+2} \cdot 2^y = 3^3$ Equating powers of 3: $x+2 = 3 \quad \text{and} \quad y = 0$ Solving for $x$: $x = 1$

Step 2: Check if the solutions are consistent

From the first equation, we found $x = 3$ and $y = 2$. From the second equation, we found $x = 1$ and $y = 0$. Since these values for $x$ and $y$ are inconsistent, let's recheck the system and work through solving them simultaneously.

Let me solve these equations more carefully.The solution to the system of equations involves logarithmic expressions that seem a bit complex for simple integer solutions. This suggests that there may be a more direct method to solve the problem without needing logs.

Let’s reanalyze the system with an assumption that the variables $x$ and $y$ are integers. Trying a trial approach with substitution might lead to cleaner results. I will now re-solve it using simpler reasoning and step-by-step integer testing.It seems like I can’t do more advanced data analysis right now. Please try again later.

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Here are 5 related questions for practice:

1. How do you solve exponential equations with different bases?
2. Can logarithms help simplify exponential equations?
3. What methods are used to solve systems of equations algebraically?
4. How can prime factorization aid in solving equations involving powers?
5. What is the relationship between exponential and logarithmic functions?

Tip: Always verify your solution by substituting the values back into the original equations.