$$ \sqrt{x-2} $$=2$$ \sqrt{y+1} $$   Equation 1                                                                                                                                                    $$ 3^{\left(z-4\right)} $$+1=$$ xy $$            Equation 2                                                                                                                                                   log(a)-1=log$$ _{y} $$(a-16) Equation 3                                                                                                                                                       $$ \int_{xx}^{a}\!y\,\mathrm{dt} $$-m.c.m($$ \text{xyza)} $$=?

To solve the given system of equations and expression, we will approach each one individually.

### Equation 1
\[ \sqrt{x-2} = 2\sqrt{y+1} \]

Square both sides to eliminate the square roots:
\[ (\sqrt{x-2})^2 = (2\sqrt{y+1})^2 \]
\[ x - 2 = 4(y + 1) \]
\[ x - 2 = 4y + 4 \]
\[ x = 4y + 6 \]
This is the relationship between $x$ and $y$.

### Equation 2
\[ 3^{(z-4)} + 1 = xy \]

### Equation 3
\[ \log(a) - 1 = \log_{y}(a - 16) \]

Convert the logarithmic equation:
\[ \log(a) - 1 = \log_{y}(a - 16) \]
\[ \log(a) - \log(10) = \log_{y}(a - 16) \]
\[ \log\left(\frac{a}{10}\right) = \log_{y}(a - 16) \]

Using the change of base formula:
\[ \frac{\log(a - 16)}{\log(y)} = \log\left(\frac{a}{10}\right) \]

### Integral and LCM Expression
\[ \int_{xx}^{a} y \, dt - \text{LCM}(xyza) \]

The integral:
\[ \int_{xx}^{a} y \, dt = y \left[t\right]_{xx}^{a} \]
\[ y(a - xx) \]

The expression becomes:
\[ y(a - xx) - \text{LCM}(xyza) \]

### Summary

1. Equation 1 yields the relationship $ x = 4y + 6 $.
2. Equation 2: $ 3^{(z-4)} + 1 = xy $.
3. Equation 3: Relates $ a $ and $ y $ through logarithms.
4. Integral and LCM: $ y(a - xx) - \text{LCM}(xyza) $.

#### Questions for Further Exploration:

1. What are the possible values for $ x $ and $ y $ from the first equation?
2. How can we solve Equation 2 for $ z $ given values of $ x $ and $ y $?
3. How does changing the base affect the logarithmic relationship in Equation 3?
4. How do we calculate the least common multiple (LCM) of $ xyza $?
5. How do different values of $ a $ affect the integral result?

#### Tip:
Always check the domain of your variables to ensure the solutions are valid within the context of the original equations.

Explore solutions to complex mathematical equations involving square roots, exponents, logarithms, integrals, and least common multiples (LCM). Learn step-by-step methods for solving equations like $ \sqrt{x-2} = 2\sqrt{y+1}, 3^{(z-4)} + 1 = xy, $ and $ \log(a) - 1 = \log_{y}(a - 16) $. Suitable for advanced high school students and anyone interested in advanced algebra and calculus problems.

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