Question

Solve the following systems of equations:

${\displaystyle \frac{2}{\sqrt{x}}+\frac{3}{\sqrt{y}}=2}$

${\displaystyle \frac{4}{\sqrt{x}}-\frac{9}{\sqrt{y}}=-1}$

Answer 1

${\displaystyle \frac{2}{\sqrt{x}}+\frac{3}{\sqrt{y}}}$ = 2

${\displaystyle \frac{4}{\sqrt{x}}-\frac{9}{\sqrt{y}}}$ = –1

Let ${\displaystyle \frac{1}{\sqrt{x}}}$ = p and ${\displaystyle \frac{1}{\sqrt{y}}}$ = q

The given equations reduce to:

2p + 3q = 2  ....(1)

4p – 9q = –1  ....(2)

Multiplying equation (1) by (3), we obtain:

6p + 9q = 6  ....(3)

Adding equation (2) and (3), we obtain:

10p = 5

p = ${\displaystyle \frac{1}{2}}$

Putting the value of p in equation (1), we obtain:

${\displaystyle 2\times \frac{1}{2}+3q=2}$

3q = 1

q = ${\displaystyle \frac{1}{3}}$

∴ p = ${\displaystyle \frac{1}{\sqrt{x}}=\frac{1}{2}}$

${\displaystyle \sqrt{x}}$ = 2

x  = 4

q = ${\displaystyle \frac{1}{\sqrt{y}}=\frac{1}{3}}$

${\displaystyle \sqrt{y}}$ = 3

y = 9

∴ x = 4, y = 9