do the same with this: xy= 4n +3, yz=5a, xz=6a +24

We are given the equations:

\[
xy = 4n + 3, \quad yz = 5a, \quad xz = 6a + 24
\]

We need to find the values of \(x\), \(y\), and \(z\), and \(yz\), with the condition that \(y\) is between \(x\) and \(z\).

### Step 1: Express \(x\) and \(z\) in terms of each other

1. Divide the first equation by the second equation to eliminate \(y\):

\[
\frac{xy}{yz} = \frac{4n + 3}{5a} \Rightarrow \frac{x}{z} = \frac{4n + 3}{5a}
\]

This implies:

\[
x = \frac{4n + 3}{5a}z
\]

2. Substitute \(x = \frac{4n + 3}{5a}z\) into the third equation \(xz = 6a + 24\):

\[
\left(\frac{4n + 3}{5a}z\right)z = 6a + 24
\]

\[
\frac{(4n + 3)z^2}{5a} = 6a + 24
\]

3. Multiply both sides by \(5a\) to eliminate the fraction:

\[
(4n + 3)z^2 = 5a(6a + 24)
\]

\[
(4n + 3)z^2 = 30a^2 + 120a
\]

### Step 2: Solve for \(z\)

\[
z^2 = \frac{30a^2 + 120a}{4n + 3}
\]

\[
z = \sqrt{\frac{30a^2 + 120a}{4n + 3}}
\]

### Step 3: Find \(x\)

Now, using \(x = \frac{4n + 3}{5a}z\):

\[
x = \frac{4n + 3}{5a} \times \sqrt{\frac{30a^2 + 120a}{4n + 3}}
\]

\[
x = \frac{\sqrt{(4n + 3)(30a^2 + 120a)}}{5a}
\]

### Step 4: Find \(y\)

Finally, using the equation \(xy = 4n + 3\):

\[
y = \frac{4n + 3}{x} = \frac{4n + 3}{\frac{\sqrt{(4n + 3)(30a^2 + 120a)}}{5a}} = \frac{5a(4n + 3)}{\sqrt{(4n + 3)(30a^2 + 120a)}}
\]

### Conclusion:

The variables are:

- \( x = \frac{\sqrt{(4n + 3)(30a^2 + 120a)}}{5a} \)
- \( z = \sqrt{\frac{30a^2 + 120a}{4n + 3}} \)
- \( y = \frac{5a(4n + 3)}{\sqrt{(4n + 3)(30a^2 + 120a)}} \)
- \( yz = 5a \)

Would you like further details or have any questions?

### Related Questions:
1. How does the presence of constants like \(n\) and \(a\) affect solving these types of algebraic equations?
2. What are the best methods to simplify radical expressions?
3. How can you interpret the relationship between variables when they are expressed as products?
4. How do you handle complex fractions in algebraic equations?
5. What steps should be taken to verify the accuracy of algebraic solutions?

### Tip:
When dealing with quadratic terms, always ensure that you simplify your expressions as much as possible before solving. This reduces the complexity and potential for errors.

Learn how to solve the system of equations xy = 4n + 3, yz = 5a, xz = 6a + 24 with a detailed step-by-step solution. This problem involves algebraic manipulation, quadratic expressions, and radical simplification. Suitable for high school students studying advanced algebra.

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