The conditions for ax+by+c=0 to be a linear equation in two variables (x,y) are
a=0 and b=0
a≠0 and b≠0
a can be zero but not b
b can be 0 but not a
Condition is that both a and b are not equal to 0. i.e a≠0 and b≠0
The equation: ax + by + c = 0 completely describes a linear equation in two variables where a, b, and c are real numbers.