Final answer:
The statement that a system of four equations in four unknowns always has a solution is false, as the solution depends on the relationship between the equations.
Step-by-step explanation:
A system of four equations in four unknowns does not always have a solution. This statement is false. There are cases where the system can have a unique solution, infinite solutions, or no solution at all. The outcome depends on the coefficients of the variables and how the equations relate to each other.
For instance, if we consider a linear system represented in matrix form as Ax = b, where A is the coefficient matrix, x is the column vector of unknowns, and b is the constant column vector, the system has a unique solution if the matrix A is invertible. However, if A is not invertible because the equations are linearly dependent (i.e., one is a linear combination of the others), the system may have either infinitely many solutions or none.