If x varies directly as y and inversely as z, and x = 14 when y = 7 and z = 2, find x when z = 4 and y =
16.

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 \( z = 4 \).

---

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.

If x varies directly as y and inversely as z, and x = 14 when y = 7 and z = 2, find x when z = 4 and y = 16.

Learn how to solve a problem involving direct and inverse variation, where x varies directly as y and inversely as z. Given x = 14 when y = 7 and z = 2, this step-by-step guide helps find x when z = 4 and y = 16. Perfect for Grades 8-10.

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