Question

Solve the following system of equations:

`2/sqrt(x) + 3/sqrt(y) = 2`

`4/sqrt(x) - 9/sqrt(y) = -1`

Answer 1

Given:

`2/sqrt(x) + 3/sqrt(y) = 2`

`4/sqrt(x) - 9/sqrt(y) = -1`

Let `u = 1/sqrt(x)` and `v = 1/sqrt(y)`.

Then the system becomes:

2u + 3v = 2

4u – 9v = –1

Multiply the first equation by 3:

6u + 9v = 6

Add this to the second equation:

(6u + 9v) + (4u – 9v) = 6 + (–1) 

⇒ 10u = 5

⇒ `u = 1/2`

Substitute `u = 1/2` into 2u + 3v = 2: 

1 + 3v = 2

⇒ 3v = 1 

⇒ `v = 1/3`

Now revert to x, y: 

`1/sqrt(x) = u = 1/2` 

⇒ `sqrt(x) = 2` 

⇒ x = 4

`1/sqrt(y) = v = 1/3` 

⇒ `sqrt(y) = 3` 

⇒ y = 9

(x, y) = (4, 9) valid since `sqrt(x)` and `sqrt(y)` are defined for these positive values.