Question

a. find the value of b so that the system has an infinite number of solutions.
b = 18
b. find a value of b so that there are no solutions to the system.
a. there are no values of b that satisfy this condition.
b. b is any real number.
c. b is any real number except 18.

Explanation:

Part b

For a system of linear equations to have no solutions, the equations must be inconsistent (parallel lines, same slope but different y - intercepts in 2 - variable linear systems). When a system has infinitely many solutions when \(b = 18\) (from part a, which implies the equations are dependent/equivalent when \(b=18\)), for no solutions, \(b\) must be a value that makes the equations inconsistent. Since the system is consistent (has infinitely many solutions) when \(b = 18\), for no solutions, \(b\) should be any real number except 18. Option A is wrong as there are values (all except 18), option B is wrong as when \(b = 18\) there are infinitely many solutions, not no solutions. So the correct option is C.

Answer:

C. \(b\) is any real number except 18.