Question

When does a system have infinitely many solutions?

a. only if there are no free variables.
b. only if there are no zero rows in row echelon form.
c. only if there is at least one non-zero row in row echelon form.
d. only if there is at least one free variable.

Answer 1

A system of linear equations has infinitely many solutions if and d. only if there are at least one free variable.

Step-by-step explanation:

A system of linear equations has infinitely many solutions if and d. only if there are at least one free variable. A free variable is a variable that can take any value, and its presence indicates that there are more unknowns than equations in the system.

In other words, the number of unknowns is larger than the number of equations, making it impossible to uniquely determine a solution.