√(x2 + 12y) + √(y2 + 12x) = 33
x + y =23
Calcular √(2x-y)

a)3
b)2
c)5
d)7
e)4

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.

√(x^2 + 12y) + √(y^2 + 12x) = 33, x + y = 23. Calcular √(2x - y): a)3 b)2 c)5 d)7 e)4

Learn how to solve the system of equations √(x^2 + 12y) + √(y^2 + 12x) = 33 and x + y = 23. This problem involves algebraic substitution and radical expressions to find √(2x - y). Suitable for high school students in Grades 9-12.

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