To solve this problem you must apply the proccedure shown below:
You have that the line is parallel to x+y=6, this means that both lines have the same slope. You can calculate the slope as below:
y=mx+b
m is the slope and b is the y-intercept
x+y=6
y=-x+6
The slope is m=-1
As the line passes through the origin, b=0, therefore, you have:
y=-x+0
y=-x
x+y=0
The answer is: x+y=0
