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

User Smisiewicz
by
7.8k points

2 Answers

2 votes

Answer:

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

User Kyle Pittman
by
8.3k points
10 votes

Answer:

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.

User Hamidreza Shokouhi
by
8.4k points

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