Question

Solve for x and y:

`35/(x + y) + 14/(x - y) = 19, 14/(x + y) + 35/(x - y) = 37`

Answer 1

Given: The system `35/(x + y) + 14/(x - y) = 19, 14/(x + y) + 35/(x - y) = 37`.

Step-wise calculation:

1. Let `a = 1/(x + y)` and `b = 1/(x - y)`.

The system becomes 35a + 14b = 19, 14a + 35b = 37.

2. Solve this 2 × 2 linear system.

Determinant D = 35 × 35 – 14 × 14

= 1225 – 196

= 1029

3. Compute a:

`a = (19 xx 35 - 14 xx 37)/D` 

= `(665 - 518)/1029` 

= `147/1029` 

= `49/343`

4. Compute b:

`b = (35 xx 37 - 19 xx 14)/D` 

= `(1295 - 266)/1029`

= `1029/1029`

= 1

5. Return to x, y:

`1/(x + y) = 49/343`

⇒ `x + y = 343/49`

= 7 

`1/(x - y) = 1` 

⇒ x – y = 1

6. Solve x + y = 7 and x – y = 1:

Add: 2x = 8 ⇒ x = 4.

Then y = 7 – x = 3.

⇒ x = 4, y = 3.