How Many Solutions

Exactly one, none at all, or infinitely many.

12 problems · about 23 minutes · A-REI.C.6, A-REI.D.11

What this lesson teaches

The student determines whether a system has one solution, no solution or infinitely many.

A linear system has exactly one solution, none, or infinitely many, and nothing else.
A row reading 0 = a non-zero number means no solution.
Free variables are unknowns without pivots, and each one adds a whole dimension of solutions.

Straightforward practice. Get the method working first.

5 problems

6 unknowns and 4 pivots. How many free variables?
Answer 2
Why 2.
What does a reduced row of [0, 0 | 7] tell you?
Answer The system is inconsistent and has no solution.
Why No solution.
A reduced row reads [0, 0 | 0]. Does it rule anything out? 1 yes, 0 no.
Answer 0
Why It says nothing at all.
A reduced row reads [0, 0 | 9]. How many solutions does the system have?
Answer 0
Why A false statement.
5 unknowns and 2 pivots. How many free variables?
Answer 3
Why 5 − 2.

The same ideas with more to keep track of.

3 problems

3 unknowns and 3 pivots with no contradiction. How many solutions?
Answer 1
Why Every unknown is pinned down.
Two lines that are the same line. How many free variables does the system have?
Answer 1
Why A whole line of solutions.
Can a linear system have exactly 2 solutions? 1 yes, 0 no.
Answer 0
Why Only three possibilities exist.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Match each reduced row to what it means.
Answer [0, 0 | 5] → No solution; [0, 0 | 0] → A redundant equation, so a free variable; [0, 1 | 3] → A pivot pinning down one unknown
Why The constant column decides between the first two.
Two parallel non-identical lines. How many solutions?
Answer 0
Why They never meet.
The Contradiction: A reduced row reads [0, 0 | 4]. How many solutions does the system have?
Answer 0
Why None.
The Count: A system has 4 unknowns and 3 pivots. How many free variables are there?
Answer 1
Why 1.