Question

Solve the following pair of linear equations:

`9/x + 2/y = 4, 12/x + 6/y = 7, x ≠ 0, y ≠ 0`

Answer 1

Given equations:

`9/x + 2/y = 4` and `12/x + 6/y = 7, x ≠ 0, y ≠ 0`

Step 1: Introduce the substitutions `u = 1/x, v = 1/y`

Then the equations become:

9u + 2v = 4   ...(1)

12u + 6v = 7   ...(2)

Step 2: Solve the system of linear equations for (u) and (v).

Multiply equation (1) by 3, to facilitate elimination:

27u + 6v = 12   ...(3)

Subtract equation (2) from equation (3):

(27u + 6v) – (12u + 6v) = 12 – 7 

15u = 5 

`u = 5/15`

`u = 1/3`

Step 3: Substitute `u = 1/3` into equation (1) to find (v): 

`9 xx 1/3 + 2v = 4` 

3 + 2v = 4 

2v = 1 

`v = 1/2`

Step 4: Recall `u = 1/x` and `v = 1/y`, 

So, `1/x = 1/3`

⇒ x = 3 

`1/y = 1/2`

⇒ y = 2