Let's break down and solve each equation step by step to analyze them individually.

Equation 1:

$\sqrt{x - 2} = 2\sqrt{y + 1}$

Square both sides to eliminate the square roots: $(\sqrt{x - 2})^2 = (2\sqrt{y + 1})^2$ $x - 2 = 4(y + 1)$ $x - 2 = 4y + 4$ $x = 4y + 6$ $\text{(Equation A)}$

Equation 2:

$3^{(z - 4)} + 1 = x \cdot y$

This equation expresses $x \cdot y$ in terms of $z$. We'll call this $\text{Equation B}$ for now.

Equation 3:

$\log_{y}(a) - 1 = \log_{y}(a - 16)$

Add 1 to both sides to make the logs more comparable: $\log_{y}(a) = \log_{y}(a - 16) + 1$

Using the properties of logarithms: $\log_{y}(a) - \log_{y}(a - 16) = 1$ $\log_{y}\left(\frac{a}{a - 16}\right) = 1$

Since $\log_{y}(y) = 1$, we have: $\frac{a}{a - 16} = y$ $a = y(a - 16)$ $a = ay - 16y$ $a(1 - y) = -16y$ $a = \frac{16y}{y - 1}$

Integral Expression:

$\int_{x^2}^{a} y \, dt - m \cdot c \cdot m(\text{xyza})$

This integral simplifies to: $y(a - x^2) - m \cdot c \cdot m(\text{xyza})$

Possibilities for the Final Answer:

Based on the equations and their interplay, let's analyze the options given: $32, 16, 2, 240, 224,$ or $14$.

1. $x = 4y + 6$
2. $x \cdot y = 3^{(z - 4)} + 1$
3. $a = \frac{16y}{y - 1}$

These values can then substitute back into the integral: $y(a - x^2) - m \cdot c \cdot m(\text{xyza})$

Estimating Possible Outcomes:

• If $x = 4y + 6$ and this matches with $x \cdot y = 3^{(z - 4)} + 1$, then potential values of $y$ and $x$ should satisfy all these relations.

Given the complexity of these interactions, $y$ values from simple integers can give hints about possible output. However, considering the values from complex functions such as logarithms and integrals, $16$, $32$, and $224$ seem like more plausible outputs from such expressions.

Given that we need all expressions to align, you can solve this by trial to check if any of these values are viable given the steps.

Would you like to see more detailed steps or perhaps delve into one of the parts further? Here are some further questions for you:

1. Should we explore specific solutions for $x$ and $y$ for Equation 1?
2. Would you like to analyze the logarithmic properties in more detail?
3. Do you want to dive deeper into solving the integral equation?
4. Should we substitute specific values into the equations to test the given options?
5. Are you interested in seeing how complex exponential equations like Equation 2 might align with logarithmic ones?

Tip: When dealing with systems of equations, substituting simple potential solutions can sometimes simplify the problem quickly.