Answer:
3
Step-by-step explanation:
[tex]\frac{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[/tex]
Answer:
3
Step-by-step explanation:
So we're given: [tex]\frac{1}{x} + \frac{1}{y} = 3[/tex] and that: [tex]xy+x+y=4[/tex]. And now we need to solve for: [tex]x^2y+xy^2[/tex].
Original equation:
[tex]\frac{1}{x} + \frac{1}{y} = 3[/tex]
Multiply both sides by xy
[tex]y+x=3xy[/tex]
Now take this and plug it as x+y into the second equation:
Original equation:
[tex]xy+x+y=4[/tex]
Substitute 3xy as x+y
[tex]xy + 3xy = 4[/tex]
Combine like terms:
[tex]4xy = 4[/tex]
Divide both sides by 4
[tex]xy=1[/tex]
Divide both sides by x:
[tex]y=\frac{1}{x}[/tex]
Original equation:
[tex]x^2y+xy^2[/tex]
Substitute 1/x as y
[tex]x^2(\frac{1}{x})+x(\frac{1}{x})^2[/tex]
Multiply values:
[tex]\frac{x^2}{x}+\frac{x}{x^2}[/tex]
Simplify:
[tex]x+\frac{1}{x}[/tex]
Substitute y as 1/x back into the equation:
[tex]x+y[/tex]
so now we just need to solve for x+y
Look back in steps to see how I got this:
[tex]y+x=3xy[/tex]
Divide both sides by 3
[tex]\frac{x+y}{3}=xy[/tex]
Original equation:
[tex]xy+x+y=4[/tex]
Substitute
[tex]\frac{x+y}{3}+x+y=4[/tex]
Multiply both sides by 3
[tex]x+y+3x+3y=12[/tex]
Combine like terms:
[tex]4x+4y=12[/tex]
Divide both sides by 4
[tex]x+y=3[/tex]
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.
