Question

If `20/(x + y) + 3/(x - y) = 7` and `8/(x - y) - 15/(x + y) = 5` then (x, y) is ______.

Options

• (–3, 5)

• (5, –3)

• (2, 3)

• (3, 2)

Answer 1

If `20/(x + y) + 3/(x - y) = 7` and `8/(x - y) - 15/(x + y) = 5` then (x, y) is (3, 2).

Explanation:

Given the system:

`20/(x + y) + 3/(x - y) = 7`

`8/(x - y) - 15/(x + y) = 5`

Let’s put:

`u = 1/(x + y)`

`v = 1/(x - y)`

Rewriting the system in terms of (u) and (v):

20u + 3v = 7

8v – 15u = 5

Now solve the two linear equations:

From the first:

20u + 3v = 7

⇒ 3v = 7 – 20u

⇒ `v = (7 - 20u)/3`

Substitute into the second:

`8((7 - 20u)/3) - 15u = 5`

⇒ `(56 - 160u)/3 - 15u = 5`

Multiply both sides by 3:

56 – 160u – 45u = 15

⇒ 56 – 205u = 15

–205u = 15 – 56

–205u = –41

⇒ `u = 41/205`

⇒ `u = 1/5`

Now substitute `(u = 1/5)` into first:

`20 xx 1/5 + 3v = 7`

⇒ 4 + 3v = 7

⇒ 3v = 3

⇒ v = 1

Recall `(u = 1/(x + y) = 1/5)` which implies (x + y = 5).

And `(v = 1/(x - y) = 1)` which implies (x – y = 1).

Solving these:

x + y = 5 

x – y = 1

Adding these:

2x = 6 

⇒ x = 3

Substitute back:

3 + y = 5

⇒ y = 2

Therefore, the solution is (x, y) = (3, 2).