Question

Write as a single rational expression a) (1/x) - (1/(x-1))b) (a-b)/(a² + (1/a))c) (3x/2y) - (5x/6y) + (x/3y)

Answer 1

a)
`(-1)/(x^2-x))`

b)
`((a^2-ab))/(a^3+1)`

c)
`(7x)/(6y)`

Explanation:

Given:

a)
`(1)/(x)-(1)/(x-1)`

now,

taking the LCM of the denominator, we have

x(x-1)

multiplying and dividing the given equation with the x(x-1)

thus,

`(x(x-1))/(x(x-1))[(1)/(x)-(1)/(x-1)]`

or

⇒
`(1)/(x(x-1))[(1*(x(x-1)))/(x)-(1*(x(x-1)))/(x-1)]`

or

⇒
`(1)/(x(x-1))[((x-1))/(1)-(x)/(1)]`

or

⇒
`(1)/(x(x-1))[x-1-x]`

or

⇒
`(x-1-x)/(x(x-1))`

or

⇒
`(-1)/(x(x-1))`

or

⇒
`(-1)/(x^2-x))`

b)
`((a-b))/(a^2+((1)/(a)))`

now,

⇒
`((a-b))/(((a^2* a+1)/(a)))`

or

⇒
`((a-b)* a)/(a^3+1)`

or

⇒
`((a^2-ab))/(a^3+1)`

c)
`(5x)/(6y)+(x)/(3y)`

taking LCM of the denominator, we have

(6y × 3y)

multiplying and dividing the given equation with the (6y × 3y)

thus,

⇒
`(6y*3y)/(6y*3y)(5x)/(6y)+(x)/(3y)`

or

⇒
`(1)/(6y*3y)((6y*3y)*5x)/(6y)+((6y*3y)x)/(3y)`

or

⇒
`(1)/(6y*3y)((3y)*5x)/(1)+((6y)x)/(1)`

or

⇒
`(3y)*5x+(6y)x)/(6y*3y)`

or

⇒
`(15xy+6xy)/(18y^2)`

or

⇒
`(21xy)/(18y^2)`

or

⇒
`(7x)/(6y)`