Question

Solve the following pair of linear equations:

`1/(2x) - 3/(4y) = 1, 4/x - 1/y = 3, x ≠ 0, y ≠ 0`

Answer 1

Given the system of equations:

`1/(2x) - 3/(4y) = 1, 4/x - 1/y = 3, x ≠ 0, y ≠ 0`

Step-wise calculation:

1. Substitute:

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

2. Rewrite the first equation:

`1/(2x) - 3/(4y) = 1`

⇒ `1/2 xx 1/x - 3/4 xx 1/y = 1`

⇒ `1/2 u - 3/4 v = 1`

3. Rewrite the second equation:

`4/x - 1/y = 3`

⇒ 4u – v = 3

4. Multiply the first rewritten equation by 4 to clear denominators:

`4 xx (1/2 u - 3/4 v) = 4 xx 1`

⇒ 2u – 3v = 4

5. Now the system is:

2u – 3v = 4   ...(i) 

4u – v = 3   ...(ii)

6. Solve (ii) for (v):

v = 4u – 3

7. Substitute in (i):

2u – 3(4u – 3) = 4 

2u – 12u + 9 = 4 

–10u + 9 = 4 

–10u = 4 – 9

–10u = –5

`u = 1/2`

8. Find (v):

`v = 4 xx 1/2 - 3`

v = 2 – 3

v = –1

9. Recall substitutions:

`u = 1/x`

`u = 1/2`

⇒ x = 2

`v = 1/y`

v = –1

⇒ y = –1

The solution to the system is x = 2, y = –1.