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
$$\frac{1}{\sqrt{x}} = \frac{1}{2} \implies \sqrt{x} = 2 \implies x = 4$$
$$\frac{1}{\sqrt{y}} = \frac{1}{3} \implies \sqrt{y} = 3 \implies y = 9$$