Question

Solve the following pair of linear equations:

`9/sqrt(x) + 2/sqrt(y) = 4, 3/sqrt(x) + 4/sqrt(y) = 3, x ≠ 0, y ≠ 0`

Answer 1

Given:

`9/sqrt(x) + 2/sqrt(y) = 4, 3/sqrt(x) + 4/sqrt(y) = 3, x ≠ 0, y ≠ 0`

Step-wise calculation:

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

Then the system becomes

9u + 2v = 4   ...(1)

3u + 4v = 3   ...(2)

2. Solve these linear equations for (u) and (v).

Multiply equation (2) by 3:

9u + 12v = 9   ...(3)

Subtract equation (1) from (3):

(9u + 12v) – (9u + 2v) = 9 – 4 

9u – 9u + 12v – 2v = 5 

10v = 5

⇒ `v = 1/2`

3. Substitute `(v = 1/2)` back into equation (2): 

`3u + 4 xx 1/2 = 3` 

3u + 2 = 3 

3u = 1

⇒ `u = 1/3`

4. Recall:

`u = 1/sqrt(x)`

`u = 1/3`

⇒ `sqrt(x) = 3`

⇒ x = 9

`v = 1/sqrt(y)`

`v = 1/2`

⇒ `sqrt(y) = 2`

⇒ y = 4