Question

If `[(x, y)][(x),(y)] = [25]` and `[(-x, y)][(2x),(y)] = [-2]`; find x and y, if:

1. x, y ∈ W (whole numbers)
2. x, y ∈ Z (integers)

Answer 1

Given: `[(x, y)][(x),(y)] = [25]`

And `[(-x, y)][(2x),(y)] = [-2]`

[x² + y²] = [25] and [–2x² + y²] = [–2]

∴ x² + y² = 25   ...(i)

And –2x² + y² = –2  ...(ii)

On subtracting, we get 3x² = 27

∴ x² = `27/3` = 9

∴ x = ±3

i. ∵ x, y ∈ W

∴ x = 3

Substituting, the value of x in (i); we have

x² + y² = 25

`\implies` (3)² + y² = 25

`\implies` 9 + y² = 25

`\implies` y² = 25 – 9 = 16

∴ y = ±4

Hence x = 3, y = 4  ...(∵ x, y ∈ W)

ii. If x, y ∈ Z

∴ x = ±3 and y = ±4