Yes, it is correct.
We can also solve this question without knowing rank, free variables,matrix , determinant etc...
We are given 3 lines.
Now , when these 3 lines lie on each other then it will give infinitely many solutions.
When at least 2 of the lines are parallel then it will give no solution.
Otherwise we will get unique solution.
Now, 3 lines will lie on each other when all 3 lines are same like x+y = 1 and 2x + 2y =2 and 3x + 3y = 3.
Now, most important thing is when 2 lines are parallel or lie on each then both lines must have the same slope or vice versa.
So, when we have a set of lines and if they all have different slopes then neither they can be parallel nor they can lie on each other.
In other words, when set of lines have different solpes then we can't get infinitely many solution or no solution.
Since , here all three lines have different slopes , so we can't get the case of infinitely many solutions or no solution.
So, by just seeing the slope, we can solve this question very quickly in case of given non-homogeneous system :)