If `xy=6,yz=10, xz=15, and x gt0`, then `xyz=`

Answer

To solve the problem step by step, we start with the given equations:

1. **Given Equations:**
- \( xy = 6 \) (1)
- \( yz = 10 \) (2)
- \( xz = 15 \) (3)
- \( x > 0 \)

2. **Expressing \( z \) in terms of \( x \) and \( y \):**
From equation (1), we can express \( y \) in terms of \( x \):
\[ y = \frac{6}{x} \]

3. **Substituting \( y \) into equation (2):**
Now, substitute \( y \) into equation (2):
\[ \left(\frac{6}{x}\right)z = 10 \]
Rearranging gives:
\[ z = \frac{10x}{6} = \frac{5x}{3} \]

4. **Substituting \( z \) into equation (3):**
Now, substitute \( z \) into equation (3):
\[ x\left(\frac{5x}{3}\right) = 15 \]
Simplifying this gives:
\[ \frac{5x^2}{3} = 15 \]

5. **Solving for \( x^2 \):**
Multiply both sides by 3 to eliminate the fraction:
\[ 5x^2 = 45 \]
Now, divide both sides by 5:
\[ x^2 = 9 \]
Taking the square root gives:
\[ x = 3 \quad (\text{since } x > 0) \]

6. **Finding \( y \):**
Now that we have \( x \), we can find \( y \):
\[ y = \frac{6}{x} = \frac{6}{3} = 2 \]

7. **Finding \( z \):**
Now we can find \( z \) using the expression we derived:
\[ z = \frac{5x}{3} = \frac{5 \cdot 3}{3} = 5 \]

8. **Calculating \( xyz \):**
Finally, we can calculate \( xyz \):
\[ xyz = x \cdot y \cdot z = 3 \cdot 2 \cdot 5 = 30 \]

Thus, the value of \( xyz \) is \( 30 \).