Question

Solve the following pair of linear equations:

`7x - 4y = 3/2 xy, x + y = 7/2 xy, x ≠ 0, y ≠ 0`

Answer 1

Given:

`7x - 4y = 3/2 xy, x + y = 7/2 xy, x ≠ 0, y ≠ 0`

Step 1: Rewrite the equations

Rewrite both equations to isolate the terms involving (x) and (y):

1. `7x - 4y = 3/2 xy`

2. `x + y = 7/2 xy`

Step 2: Divide both sides by (xy ≠ 0) to introduce new variables

Let:

`u = 1/x, v = 1/y`

Rewrite equations by dividing all terms by (xy):

For (1):

`(7x)/(xy) - (4y)/(xy) = 3/2`

⇒ `7 1/y - 4 1/x = 3/2`

⇒ `7v - 4u = 3/2`

For (2):

`x/(xy) + y/(xy) = 7/2`

⇒ `1/y + 1/x = 7/2`

⇒ `v + u = 7/2`

Step 3: Set up the system for (u) and (v)

`7⁢v - 4u = 3/2` 

`u + v = 7/2`

Step 4: Solve the system for (u) and (v)

From the second equation:

`v = 7/2 - u`

Substitute into the first:

`7(7/2 - u) - 4u = 3/2`

Simplify:

`49/2 - 7u - 4u = 3/2`

⇒ `49/2 - 11u = 3/2`

Bring constants together:

`-11u = 3/2 - 49/2`

`-11u = -46/2`

–11u = – 23 

So, `u = 23/11`.

Substitute back for (v):

`v = 7/2 - 23/11`

`v = 77/22 - 46/22`

`v = 31/22`

Step 5: Find (x) and (y) from (u) and (v)

Recall:

`u = 1/x`

`u = 23/11`

⇒ `x = 11/23`

`v = 1/y`

`v = 31/22`

⇒ `y = 22/31`