How Many Solutions

Exactly one, none at all, or infinitely many.

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

NameDate

Figure — use these to answer the problems

Make the second row a multiple of the first, including the constant. The system then has infinitely many solutions.

row 1, x1

row 1, y2

row 1, constant3

row 2, x2

row 2, y4

row 2, constant7

Figure 1

Straightforward practice. Get the method working first.

5 problems

6 unknowns and 4 pivots. How many free variables?
Answer
2
What does a reduced row of [0, 0 | 7] tell you?
- The system is inconsistent and has no solution.
- The system has infinitely many solutions.
A reduced row reads [0, 0 | 0]. Does it rule anything out? 1 yes, 0 no.
Answer
A reduced row reads [0, 0 | 9]. How many solutions does the system have?
Answer
5 unknowns and 2 pivots. How many free variables?
Answer

The same ideas with more to keep track of.

3 problems

3 unknowns and 3 pivots with no contradiction. How many solutions?
Answer
Two lines that are the same line. How many free variables does the system have?
Answer
Can a linear system have exactly 2 solutions? 1 yes, 0 no.
Answer

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

4 problems

9
Match each reduced row to what it means.
Draw a line from each item on the left to its match on the right.
- [0, 0 | 5]
- [0, 0 | 0]
- [0, 1 | 3]
- No solution
- A redundant equation, so a free variable
- A pivot pinning down one unknown
Two parallel non-identical lines. How many solutions?
Answer
The Contradiction
A reduced row reads [0, 0 | 4]. How many solutions does the system have?
Answer
The Count
A system has 4 unknowns and 3 pivots. How many free variables are there?
Answer