Question

How do you solve the rational equation `5 / (y - 2) = y + 2`?

Answer 1

`color(magenta)(therefore y=3 or y=-3`

Explanation:

`5/(y-2)=y+2`

`(y+2)(y-2)=5`

Identity:
`color(red)((a+b)(a-b)=a^2-b^2`

`y^2-4=5`

`y^2-4-5=0`

`y^2-9=0`

Identity:
`color(red)((a+b)(a-b)=a^2-b^2`

`(y+3)(y-3)=0`

`therefore y-3=0 or y+3=0`

`color(magenta)(therefore y=3 or y=-3`

Answer 2

`y=+-3`

Explanation:

`"multiply both sides by "(y-2)`

`cancel((y-2))xx5/cancel((y-2))=(y-2)(y+2)`

`rArr(y-2)(y+2)=5larrcolor(blue)"expand factors"`

`rArry^2-4=5larrcolor(blue)"add 4 to both sides"`

`rArry^2=9`

`color(blue)"take square root of both sides"`

`rArry=+-3larrcolor(blue)"note plus or minus"`

`color(blue)"As a check"`

`y=3to5/(3-2)=3+2to5=5" True"`

`y=-3to5/(-3-2)=-3+2to-1=-1" True"`

`rArry=+-3" are the solutions"`