Question

Given that 1/x+1/y=3 and xy+x+y=4, compute x^2y+xy^2.

Answer 1

3

Explanation:

`(y+x)/(xy) =3\\x+y=3xy\\xy+x+y=4\\xy+3xy=4\\4xy=4\\xy=1\\x^2y+xy^2=xy(x+y)=xy(3xy)=3(xy)^2=3(1)^2=3`

Answer 2

3

Explanation:

So we're given:
`(1)/(x) + (1)/(y) = 3` and that:
`xy+x+y=4`. And now we need to solve for:
`x^2y+xy^2`.

Original equation:

`(1)/(x) + (1)/(y) = 3`

Multiply both sides by xy

`y+x=3xy`

Now take this and plug it as x+y into the second equation:

Original equation:

`xy+x+y=4`

Substitute 3xy as x+y

`xy + 3xy = 4`

Combine like terms:

`4xy = 4`

Divide both sides by 4

`xy=1`

Divide both sides by x:

`y=(1)/(x)`

Original equation:

`x^2y+xy^2`

Substitute 1/x as y

`x^2((1)/(x))+x((1)/(x))^2`

Multiply values:

`(x^2)/(x)+(x)/(x^2)`

Simplify:

`x+(1)/(x)`

Substitute y as 1/x back into the equation:

`x+y`

so now we just need to solve for x+y

Look back in steps to see how I got this:

`y+x=3xy`

Divide both sides by 3

`(x+y)/(3)=xy`

Original equation:

`xy+x+y=4`

Substitute

`(x+y)/(3)+x+y=4`

Multiply both sides by 3

`x+y+3x+3y=12`

Combine like terms:

`4x+4y=12`

Divide both sides by 4

`x+y=3`

So now we finally arrive to our solution 3!!!!! I swear I felt like I was going in circles, and I was about to just stop trying to solve, because I had no idea what I was doing, sorry if I made some unnecessary intermediate steps.