The problem states that $x$ varies directly as $y$ and inversely as $z$, which can be written as:

$x = k \frac{y}{z}$

where $k$ is the constant of proportionality.

Step 1: Find the constant $k$

We are given that $x = 14$, $y = 7$, and $z = 2$. Substituting these values into the equation:

$14 = k \frac{7}{2}$

Solving for $k$:

$14 = k \times 3.5 \implies k = \frac{14}{3.5} = 4$

So, the constant $k = 4$.

Step 2: Find $x$ when $y = 16$ and $z = 4$

Now that we have $k = 4$, we substitute $y = 16$ and $z = 4$ into the equation:

$x = 4 \frac{16}{4} = 4 \times 4 = 16$

Thus, $x = 16$ when $y = 16$ and $z = 4$.

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Would you like more details or further clarification?

Here are 5 related questions for practice:

1. If $x$ varies directly as $y^2$ and inversely as $z$, how would the formula change?
2. What would happen to $x$ if $z$ were doubled and $y$ stayed the same?
3. How would you find $x$ if $z = 0$?
4. If $y = 12$ and $z = 3$, and $k = 5$, what is $x$?
5. How can you interpret direct and inverse variation in terms of graphs?

Tip: Inverse variation means that as one variable increases, the other decreases, while direct variation means both increase or decrease together.