If `xy=22,yz=10, xz=55, and x gt 0`, then `xyz=`

Answer

To solve the problem step by step, we need to find the value of \( xyz \) given the equations \( xy = 22 \), \( yz = 10 \), and \( xz = 55 \) with the condition that \( x > 0 \).

### Step 1: Express variables in terms of one variable
We can express \( x \), \( y \), and \( z \) in terms of \( z \).

From \( xz = 55 \):
\[ x = \frac{55}{z} \]

From \( yz = 10 \):
\[ y = \frac{10}{z} \]

### Step 2: Substitute into the first equation
Now, substitute the expressions for \( x \) and \( y \) into the equation \( xy = 22 \):
\[ \left(\frac{55}{z}\right) \left(\frac{10}{z}\right) = 22 \]

### Step 3: Simplify the equation
This simplifies to:
\[ \frac{55 \times 10}{z^2} = 22 \]
\[ \frac{550}{z^2} = 22 \]

### Step 4: Solve for \( z^2 \)
Now, multiply both sides by \( z^2 \) to eliminate the fraction:
\[ 550 = 22z^2 \]

Next, divide both sides by 22:
\[ z^2 = \frac{550}{22} \]
\[ z^2 = 25 \]

### Step 5: Find \( z \)
Taking the square root of both sides, we find:
\[ z = 5 \quad (\text{since } z > 0) \]

### Step 6: Find \( x \) and \( y \)
Now that we have \( z \), we can find \( x \) and \( y \):
\[ x = \frac{55}{z} = \frac{55}{5} = 11 \]
\[ y = \frac{10}{z} = \frac{10}{5} = 2 \]

### Step 7: Calculate \( xyz \)
Now, we can calculate \( xyz \):
\[ xyz = x \cdot y \cdot z = 11 \cdot 2 \cdot 5 \]
\[ xyz = 110 \]

Thus, the value of \( xyz \) is \( \boxed{110} \).

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