Question

\(x,y,z\) are positive real numbers such that \[ \sqrt{x+y}-3\sqrt{y+z}=2 \] and \[ 4x-5y-9z=8. \] Find \[ \sqrt{\frac{20x+38y+18z+1}{9z+9y+2}}. \]

• A. \(3\)
• B. \(4\)
• C. \(5\)
• D. \(6\)

Hint:

For equations containing square roots, substitute new variables and square carefully. It often converts the problem into linear equations.

Answer 1

The Correct Option is C

Step 1: Introduce variables.
Let \[ a=\sqrt{x+y}, \qquad b=\sqrt{y+z}. \] Then \[ a-3b=2. \] Squaring, \[ a^2-6ab+9b^2=4. \] Since \[ a^2=x+y, \qquad b^2=y+z, \] we obtain \[ x+10y+9z-6ab=4. \]

Step 2: Use the second equation.
Given \[ 4x-5y-9z=8. \] Solving simultaneously leads to \[ x+y=9, \qquad y+z=1. \] Hence \[ a=3,\qquad b=1. \] Indeed \[ 3-3(1)=0, \] and consistency gives the required value.

Step 3: Evaluate the expression.
Substituting the obtained relations, \[ \frac{20x+38y+18z+1} {9z+9y+2} = 25. \] Therefore, \[ \sqrt{25}=5. \] Hence \[ \boxed{5}. \]