Question

`7/(z+1) - z - 5/(z^2-1) = 6/z` ? P.S. I have no idea on how to solve this problem.

Answer 1

Given:`" "7/(z+1)-z-5/(z^2-1)=6/z`

We need to find a common denominator

Observe that `z^2-1` is the same as `z^2-1^2=(z-1)(z+1)`.
So I choose the common denominator to be `(z-1)(z+1)`

We need to 'force' each fraction to have this as its denominator.

So we start with:
`7/(z+1)-z-5/((z-1)(z+1))=6/z`

Now we start to change things. I am not changing the RHS side as I can 'get rid' of the z denominator by cross multiplying.

`color(green)([7/(z+1)color(red)(xx1)] -[zcolor(red)(xx1)]-[5/((z-1)(z+1))]=[6/z] )`

`color(white)(.)`
The equation gets folded over 2 lines due to its length

`color(green)([7/(z+1)color(red)(xx(z-1)/(z-1))] -[zcolor(red)(xx((z-1)(z+1))/((z-1)(z+1)))]-[5/((z-1)(z+1))] =[6/z]`

`color(green)((7(z-1)-z(z-1)(z+1)-5)/((z-1)(z+1))=6/z )`

Multiply both sides by `z`

`color(green)((7z(z-1)-z^2(z-1)(z+1)-5z)/((z-1)(z+1))=6 )`

Multiply both sides by `(z-1)(z+1)`

`color(green)(7z(z-1)-z^2(z-1)(z+1)-5z=6(z-1)(z+1)`

`7z^2-7z-z^4+z^2-5z =6z^2-6`

Hence

`z^4-2z^2+12z-6=0`

I do not know how to take it on from this point!
I will ask another contributor I know to take a look!

Answer 2

`z=3` for corrected question.

Explanation:

I think the equation in the question should be:

`7/(z+1)-(z-5)/(z^2-1)=6/z`

Note that `z^2-1 = (z-1)(z+1)`

So in order to change this rational equation into a polynomial one, we can multiply by `z(z-1)(z+1)` to get:

`7z(z-1)-z(z-5) = 6(z^2-1)`

which multiplies out to give:

`color(red)(cancel(color(black)(7z^2)))-7z-color(red)(cancel(color(black)(z^2)))+5z = color(red)(cancel(color(black)(6z^2)))-6`

The terms in `z^2` cancel out and we can combine the remaining terms to get:

`-2z = -6`

Divide both sides by `-2` to get:

`z = 3`

Finally we need to check that this is a valid solution by making sure that none of the denominators in the original equation are `0` when we substitute this value of `z`. No problem.