Let's see an intuitive approach to solve this problem i.e. I'll not use the concept of rank here.
Given, m linear equation with n variables. This means we have a m*n matrix.
Case1: m<n: Let we a take a 2*3 matrix. In this case, we can't the fill the whole space as no of rows < no of columns. If it would've been same, then columns should be linearly independent to always have a solution. So, this statement is False.
Case2: m>n: Let we take a 3*2 matrix. In this case, if b is linear combination of columns of matrix A, then there would be a solution. So, we can't say here none of the system have a solution. So, this statement is False.
Case3: m=n: Let we take a 3*3 matrix. In this case we have no of columns = no of rows. If we have n LI columns, then there is always a solution. So, this statement is true.
PS: Working approach in these types of questions is simple. We just need to care of "Quantifiers." Keywords like "ALL", "At least", "Always", "NEVER" can be used to interpret the behavior of system and we can solve using the intuition.