Problem Statement
Solve the pair of equations for $x$ and $y$:
$\frac{2}{\sqrt{x}} + \frac{3}{\sqrt{y}} = 2$
$\frac{4}{\sqrt{x}} - \frac{9}{\sqrt{y}} = -1$
Verified Solution & Marking Scheme
Substitute u = 1/√x and v = 1/√y
Let $u = \frac{1}{\sqrt{x}}$ and $v = \frac{1}{\sqrt{y}}$:
$2u + 3v = 2$ ...(1)
$4u - 9v = -1$ ...(2)
Solve the Linear System for u and v
Multiply equation (1) by 3:
$6u + 9v = 6$
Add to equation (2):
$(6u + 9v) + (4u - 9v) = 6 + (-1) \implies 10u = 5 \implies u = \frac{1}{2}$
Substitute $u = 1/2$ into (1):
$2(1/2) + 3v = 2 \implies 1 + 3v = 2 \implies 3v = 1 \implies v = \frac{1}{3}$
Solve for x and y
$\frac{1}{\sqrt{x}} = \frac{1}{2} \implies \sqrt{x} = 2 \implies x = 4$
$\frac{1}{\sqrt{y}} = \frac{1}{3} \implies \sqrt{y} = 3 \implies y = 9$