Question

Solve the following pair of linear equations:

7x + 4y = 6xy, x + 3y = – 4xy, x ≠ 0, y ≠ 0

Answer 1

Given:

7x + 4y = 6xy, x + 3y = – 4xy with the condition x ≠ 0 and y ≠ 0.

Step 1: Rewrite each equation by dividing both sides by (xy):

Since (x ≠ 0) and (y ≠ 0), divide each term by (xy):

1. For 7x + 4y = 6xy:

`(7x)/(xy) + (4y)/(xy) = (6xy)/(xy)`

⇒ `7/y + 4/x = 6`

2. For x + 3y = –4xy:

`x/(xy) + (3y)/(xy) = (-4xy)/(xy)`

⇒ `1/y + 3/x = -4`

Step 2: Introduce substitution variables

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

Rewrite the equations in terms of (u) and (v):

7v + 4u = 6 

v + 3u = –4

Step 3: Solve the linear system in (u, v)

From the second equation:

v = –4 – 3u

Substitute into the first equation:

7(–4 – 3u) + 4u = 6 

–28 – 21u + 4u = 6 

–28 – 17u = 6 

–17u = 34 

u = –2

Step 4: Find (v)

v = –4 – 3(–2)

v = –4 + 6

v = 2

Step 5: Back-substitute to find (x) and (y)

`u = 1/x`

u = –2

⇒ `x = -1/2`

`v = 1/y`

v = 2

⇒ `y = 1/2`

The solution to the system is `x = -1/2, y = 1/2`.