Question

Solve for x and y:

`1/(2(x + 2y)) + 5/(3(3x - 2y)) = (-3)/2, 5/(4(x + 2y)) - 3/(5(3x - 2y)) = 61/60`

Answer 1

Given: `1/(2(x + 2y)) + 5/(3(3x - 2y)) = (-3)/2, 5/(4(x + 2y)) - 3/(5(3x - 2y)) = 61/60`

Step-wise calculation:

1. Set u = x + 2y and v = 3x – 2y.

Then `1/u` and `1/v` simplify the rational system.

Let `a = 1/u, b = 1/v`.

2. Rewrite the two equations in a and b:

`(1) (1/2)a + (5/3)b = -3/2`

`(2) (5/4)a - (3/5)b = 61/60`

3. Clear denominators:

Multiply (1) by 6

⇒ 3a + 10b = –9

Multiply (2) by 60

⇒ 75a – 36b = 61

So, the linear system is 3a + 10b = –9, 75a – 36b = 61.

4. Eliminate b: multiply the first equation by 36

⇒ 108a + 360b = –324

Multiply the second by 10

⇒ 750a – 360b = 610

Add these two: 858a = 286

⇒ `a = 286/858 = 1/3`

5. Substitute `a = 1/3` into 3a + 10b = –9:

`3(1/3) + 10b = -9`

⇒ 1 + 10b = –9

⇒ 10b = –10

⇒ b = –1

6. Recover u and v:

`a = 1/u = 1/3` 

⇒ u = 3

`b = 1/v = -1` 

⇒ v = –1

7. Solve for x and y from u and v:

u = x + 2y = 3

v = 3x – 2y = –1

Add the two equations:

(x + 2y) + (3x − 2y) = 3 + (–1) 

⇒ 4x = 2

⇒ `x = 1/2` 

Plug x into x + 2y = 3:

`1/2 + 2y = 3`

⇒ `2y = 5/2`

⇒ `y = 5/4`