Given the system of equations:

\[
xy = 5n
\]
\[
yz = 2n
\]
\[
xz = 91
\]

Let's solve for \( x \), \( y \), and \( z \) in terms of \( n \).

### Step 1: Express \( y \) in terms of \( x \) and \( n \)

From the equation \( xy = 5n \), we can solve for \( y \) as:

\[
y = \frac{5n}{x}
\]

### Step 2: Express \( y \) in terms of \( z \) and \( n \)

From the equation \( yz = 2n \), we can solve for \( y \) again as:

\[
y = \frac{2n}{z}
\]

### Step 3: Set the two expressions for \( y \) equal

Now that we have two expressions for \( y \), we can set them equal to each other:

\[
\frac{5n}{x} = \frac{2n}{z}
\]

Canceling out \( n \) from both sides:

\[
\frac{5}{x} = \frac{2}{z}
\]

### Step 4: Solve for \( z \) in terms of \( x \)

Cross-multiply to get:

\[
5z = 2x
\]

Solving for \( z \):

\[
z = \frac{2x}{5}
\]

### Step 5: Substitute into the third equation

Now, substitute \( z = \frac{2x}{5} \) into the third equation \( xz = 91 \):

\[
x \left( \frac{2x}{5} \right) = 91
\]

This simplifies to:

\[
\frac{2x^2}{5} = 91
\]

Multiply both sides by 5:

\[
2x^2 = 455
\]

Now divide by 2:

\[
x^2 = 227.5
\]

So,

\[
x = \sqrt{227.5} \approx 15.08
\]

### Step 6: Solve for \( z \)

Using the expression \( z = \frac{2x}{5} \):

\[
z = \frac{2(15.08)}{5} \approx 6.03
\]

### Step 7: Solve for \( y \)

Using the expression \( y = \frac{5n}{x} \), and assuming \( n = 1 \) for simplicity:

\[
y = \frac{5}{15.08} \approx 0.33
\]

Thus, the approximate values are:
- \( x \approx 15.08 \)
- \( y \approx 0.33 \)
- \( z \approx 6.03 \)

Would you like more details on any specific part of this solution, or clarification?

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Here are 5 related questions to expand your understanding:

1. How would the solution change if \( n \) is given a specific value, like \( n = 2 \)?
2. Can you verify these values by substituting them back into the original equations?
3. How would you approach this system if it involved inequalities instead of equalities?
4. What if one of the variables is zero, does the system still hold valid?
5. How can you interpret these values geometrically if the variables represent dimensions?

**Tip:** When solving systems of equations, try eliminating one variable at a time by substitution or setting up equal expressions to simplify the system.

Learn how to solve the system of equations xy = 5n, yz = 2n, xz = 91 with step-by-step solutions and detailed calculations. This problem involves using algebraic methods to find the values of x, y, and z in terms of n. Suitable for high school students studying algebra.

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