Question

How do you solve the linear equation system `(4/x)-(3/y)=1` & `(6/x)-(3/y)=-4`?

Answer 1

`(x,y)=(-2/5,-3/11)`

Explanation:

`4x-3y=1`
`color(white)("XXX")rarr 4y-3x=xy` (after multiplying everything by `xy`)

`color(white)("XXX")rarr 4y-xy=3x`

`color(white)("XXX")rarr y(4-x) = 3x`

[1]`color(white)("XXX")rarr y=(3x)/(4-x)`

`6/x-3/y=-4`
`color(white)("XXX")rarr 6y-3x=-4xy`

`color(white)("XXX")rarr 6y+4xy=3x`

`color(white)("XXX")rarr y(6+4x)=3x`

[2]`color(white)("XXX")rarr y=(3x)/(6+4x)`

Combining [1] and [2]
`color(white)("XXX")(3x)/(4-x)=y=(3x)/(6+4x)`

`color(white)("XXX")rarr 4-x=6+4x`

`color(white)("XXX")rarr -5x = 2`

`color(white)("XXX")rarr x= -2/5`

Substituting `(-2/5)` for `x` in [1]
`color(white)("XXX")y = (3*(-2/5))/(4-(-2/5))`

`color(white)("XXXX")=(-6/5)/((20+2)/5)`

`color(white)("XXXX")=-6/22 = -3/11`

(Substitute these values back into the original equations to verify).