A linear system has exactly one solution, no solution, or infinitely many. There is never any other possibility.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
One solution
Every unknown has a pivot. In two dimensions this is two lines crossing at a single point.
No solution
A row reduces to 0 = 5, which is false. Geometrically the lines are parallel and never meet.
Infinitely many
A row reduces to 0 = 0, which says nothing. Geometrically the two lines are the same line.
Free variables
An unknown without a pivot is free. It may take any value, and each choice gives another solution.
Counting
Free variables equal unknowns minus pivots. One free variable gives a line of solutions; two give a plane.
Exactly three possibilities
A linear system has one solution, none, or infinitely many. It can never have exactly two, which follows from the fact that the average of two solutions is another solution.
Recognising no solution
A row reading 0 = nonzero is an impossible equation, so the system is inconsistent. That row is the signature, and it appears during elimination rather than at the end.
Free variables give infinitely many
A column without a pivot corresponds to a free variable, which can take any value. Each free variable adds a dimension to the solution set, described parametrically.
The geometry of each case
Three planes can meet at a point, along a line, in a plane, or not at all. The algebraic cases correspond exactly to those geometric configurations.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Malik Read a Contradiction
Malik reduces a system and reaches the row [0, 0 | 5].
- Step 1
He translates the row back into an equation: 0x + 0y = 5.
Step 4: Your Turn
Practice makes it stick.
The Contradiction
Problem 1 of 2
A reduced row reads [0, 0 | 4]. How many solutions does the system have?
The Count
Problem 2 of 2
A system has 4 unknowns and 3 pivots. How many free variables are there?
Count the Solutions
1 of 8
A reduced row reads [0, 0 | 0]. Does it rule anything out? 1 yes, 0 no.
2 of 8
A reduced row reads [0, 0 | 9]. How many solutions does the system have?
3 of 8
5 unknowns and 2 pivots. How many free variables?
4 of 8
3 unknowns and 3 pivots with no contradiction. How many solutions?
5 of 8
Two lines that are the same line. How many free variables does the system have?
6 of 8
Can a linear system have exactly 2 solutions? 1 yes, 0 no.
7 of 8
Match each reduced row to what it means.
Tap a card on the left to start.
8 of 8
Two parallel non-identical lines. How many solutions?
Step 5: Quick Check
Show what you know.
Question 1 of 2
6 unknowns and 4 pivots. How many free variables?
Question 2 of 2
What does a reduced row of [0, 0 | 7] tell you?
What You Learned
- 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.