We are given the system of equations:

1. $\sqrt{x^2 + 12y} + \sqrt{y^2 + 12x} = 33$
2. $x + y = 23$

We are asked to find $\sqrt{2x - y}$.

Step 1: Solve for $x$ and $y$

From the second equation, $x + y = 23$, we can express $y$ in terms of $x$:

$y = 23 - x$

Step 2: Substitute $y = 23 - x$ into the first equation

Substitute $y = 23 - x$ into the first equation:

$\sqrt{x^2 + 12(23 - x)} + \sqrt{(23 - x)^2 + 12x} = 33$

Simplify both expressions under the square roots:

$\sqrt{x^2 + 276 - 12x} + \sqrt{(23 - x)^2 + 12x} = 33$

For $(23 - x)^2$, expand it:

$(23 - x)^2 = 529 - 46x + x^2$

Substitute this into the second square root:

$\sqrt{x^2 + 276 - 12x} + \sqrt{529 - 46x + x^2 + 12x} = 33$

Simplify the second square root:

$\sqrt{x^2 + 276 - 12x} + \sqrt{529 - 34x + x^2} = 33$

Step 3: Solve the system numerically

Solving this system of equations by substitution or trial and error will help us find $x$ and $y$. Once we find $x$ and $y$, we can use them to find $\sqrt{2x - y}$.

Let's calculate the approximate values of $x$ and $y$.The solutions for $x$ are $x = 10$ and $x = 13$.

Step 4: Find the corresponding values of $y$

From $x + y = 23$, we can calculate $y$:

• If $x = 10$, then $y = 23 - 10 = 13$.
• If $x = 13$, then $y = 23 - 13 = 10$.

Step 5: Calculate $\sqrt{2x - y}$

Now we will compute $\sqrt{2x - y}$ for both pairs $(x, y) = (10, 13)$ and $(x, y) = (13, 10)$:

1. For $x = 10$ and $y = 13$: $\sqrt{2x - y} = \sqrt{2(10) - 13} = \sqrt{20 - 13} = \sqrt{7}$

2. For $x = 13$ and $y = 10$: $\sqrt{2x - y} = \sqrt{2(13) - 10} = \sqrt{26 - 10} = \sqrt{16} = 4$

Conclusion

The correct value of $\sqrt{2x - y}$ is $\boxed{4}$.

Let me know if you want further details or have any questions!

---

Here are 5 related questions for further practice:

1. Solve $\sqrt{x^2 + 15y} + \sqrt{y^2 + 15x} = 40$ and $x + y = 30$.
2. Find the value of $\sqrt{x + 2y}$ given $x + y = 25$ and $x^2 + y^2 = 650$.
3. Solve the system $\sqrt{x^2 + 3y} + \sqrt{y^2 + 3x} = 15$ and $x + y = 10$.
4. If $x + y = 20$ and $\sqrt{x^2 + y^2 + xy} = 30$, find $\sqrt{2x - y}$.
5. Solve $\sqrt{x^2 + 5y} + \sqrt{y^2 + 5x} = 20$ and $x + y = 15$.

Tip: When solving equations with square roots, isolating one root at a time can simplify the process.