If x, y, and z are positive with xy=24, xz=48, and yz=72 , what is the value of $x+y+z$?

We are given the equations:

\[
xy = 24, \quad xz = 48, \quad yz = 72
\]

We want to find the value of $x + y + z$.

### Step 1: Multiply all three given equations together.

\[
(xy) \cdot (xz) \cdot (yz) = 24 \cdot 48 \cdot 72
\]

The left-hand side becomes:

\[
(x^2y^2z^2) = (xyz)^2
\]

Thus, we have:

\[
(xyz)^2 = 24 \cdot 48 \cdot 72
\]

### Step 2: Simplify the product on the right-hand side.

Let's compute $24 \cdot 48 \cdot 72$:

\[
24 \cdot 48 = 1152
\]
\[
1152 \cdot 72 = 82944
\]

Thus, we have:

\[
(xyz)^2 = 82944
\]

### Step 3: Take the square root of both sides.

\[
xyz = \sqrt{82944} = 288
\]

### Step 4: Use this result to find individual variables.

Now we know:

\[
xyz = 288
\]

We also know the three original equations: $xy = 24$, $xz = 48$, and $yz = 72$.

To find $x$, $y$, and $z$, divide the product $xyz$ by each of the given pairwise products:

\[
x = \frac{xyz}{yz} = \frac{288}{72} = 4
\]
\[
y = \frac{xyz}{xz} = \frac{288}{48} = 6
\]
\[
z = \frac{xyz}{xy} = \frac{288}{24} = 12
\]

### Step 5: Calculate $x + y + z$.

\[
x + y + z = 4 + 6 + 12 = 22
\]

Thus, the value of $x + y + z$ is $ \boxed{22} $.

---

Would you like more details or have any questions?

Here are 5 related questions:
1. What would happen if we changed the product values slightly, like $xy = 36$?
2. How would this problem change if one of $x, y, z$ was negative?
3. Can we solve this system geometrically or using matrices?
4. What if we only knew two of the three products, could we still determine $x + y + z$?
5. How does knowing $x + y + z$ help in optimizing related functions?

**Tip:** When solving for multiple variables, try multiplying equations to reduce the number of unknowns.

If x, y, and z are positive with xy=24, xz=48, and yz=72, what is the value of x + y + z?

Discover how to solve for x + y + z given the equations xy=24, xz=48, and yz=72. Follow a step-by-step process that includes multiplying equations, applying square roots, and isolating variables. This problem covers key algebraic techniques, suitable for students in Grades 9-10.

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