Question

A system of linear equations has 2 equations in 5 unknowns. which one of the following statements is true?

a. if the system has a solution, the solution must be unique.
b. the system always has a solution.
c. if the system has a solution, the family of solutions must have at least 3 parameters.
d. if the system has a solution, the family of solutions must have exactly 3 parameters.
e. the system never has a solution.

Answer 1

The system with 2 equations in 5 unknowns may have infinitely many solutions if it is consistent. The correct statement is 'if the system has a solution, the family of solutions must have at least 3 parameters' because the solution space can be described by at least three free variables. Option c.

Step-by-step explanation:

In the given scenario, a system of linear equations has 2 equations in 5 unknowns.

According to linear algebra, if the number of unknowns exceeds the number of equations, the system does not have enough constraints to determine a unique solution. In this case, the system may have infinitely many solutions, depending on the consistency of the equations.

Therefore, statement (c) 'if the system has a solution, the family of solutions must have at least 3 parameters' is true. This implies that the solution space can be described by at least three free variables, or parameters, which define a family of solutions.

So Option c.