Help me graph manually, for level sets

[seto6]

Homework Statement

Characterize and Sketch several level sets for the function::

z=x²+y²/2(x+y)

Homework Equations

N/A

The Attempt at a Solution

i tried to set a z, for example say for z=0.5, then i get in the form of::

x+y = x²+y²

now its the difficult part, graphing it, i know doing on the computer is easy, but i want to know how to do it by hand, any tricks or tips that will make it easier to do so, rather than point by point.

same problem with z=k

2k(x+y)=x²+y²

 

[Mark44]

Homework Statement

Characterize and Sketch several level sets for the function::

z=x²+y²/2(x+y)

What you have written is ambiguous, but I think you mean
$$z = \frac{x^2 + y^2}{2(x + y)}$$

You can click the equation I wrote to see my LaTeX script.

If you write it without using LaTeX, use parentheses to separate the numerator and denominator, like so:
z = (x² + y²)/(2(x + y))

Homework Equations

N/A

The Attempt at a Solution

i tried to set a z, for example say for z=0.5, then i get in the form of::

x+y = x²+y²

now its the difficult part, graphing it, i know doing on the computer is easy, but i want to know how to do it by hand, any tricks or tips that will make it easier to do so, rather than point by point.

The equation above is equivalent to
x² - x + y² - y = 0.
Complete the square in the x and y terms to see that the level curve for z = .5 is a circle.

Be advised that your equation above is not equivalent to the one you started with, since the original function is undefined if x + y = 0.

same problem with z=k

2k(x+y)=x²+y²

Same thing as above should work.

 

[seto6]

got it thank you