MATH 141-501 Lecture Notes
Section 2.3
Recall: given any system of linear equations, there are 3 distinct possibilities:
1. The system has exactly 1 solution.
2. The system has infinitely many solutions.
3. The system has no solution.
Question: What do these possibilities look like, in terms of matrices?
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Example: System with One Solution
The system
x + 51y = 60
−x + y = 980
has one solution: x = −960, y = 20.
The augmented matrix for the system is
1 51 60
.
−1 1 980
The row-reduced form of the augmented matrix (rref) is :
1 0 960
.
0 1 20
Remark: A system has a unique solution when the row reduced form of its
augmented matrix has
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Example: System with Infinitely Many Solutions
The system
5x + 10y = 1
1
1
x+y =
2
10
has infinitely many solutions.
The augmented matrix for the system is
5 10 1
1
1
1 10
2
The row-reduced form of the augmented matrix (rref) is:
1 2 0.2
.
0 0 0
Remark: A system has infinitely many solutions when the row reduced
form of its augmented matrix has
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Finding Parametric Forms When System Has Infinitely Many
Solutions
Solve the following system of equations:
x + 2y − 3z + w = −2
3x − y − 2z − 4w = 1
2x + 3y − 5z + w = 1
Step 1: Write the system as an

1 2
 3 −1
2 3
augmented matrix.

−3 1 −2
−2 −4 1 
−5 1
1
Step 2: Find the row-reduced form (rref) of the augmented matrix.
Step 3: Translate the row-reduced form back into a system of equations.
Step 4: Now, solve for some variables in terms of the others. (“The others”
are called
and will be represented
Remark: Indvidual solutions of the system may be found by
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Example: A system with no solution.
The system
3x − y = 10
9x − 3y = 110
has no solution.
The augmented matrix for this system is
3 −1 10
9 −3 110
The row-reduced form of the augmented matrix is
3 −1 10
0 0 80
Remark: A system has no solution when the row-reduced form of its augmented matrix contains a row which has
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Number of Solutions
Recall: given any system of linear equations, there are 3 distinct possibilities:
1. The system has exactly 1 solution.
2. The system has infinitely many solutions.
3. The system has no solution.
If there are at least as many variables in the system as equations, then
If there are fewer variables in the system than equations, then
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Example: (p. 104, Finite Mathematics for the Managerial, Life, and Social
Sciences, 11th edition).
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Example, continued
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