special cases system of linear equations

1. Special Cases of Systems of Linear Equations
Inconsistent (Contradictory) Systems
{x + y = 2
x + y = 3
E1 – E2: x + y = 2
) x + y = 3
0 = –1
These systems are called inconsistent or contradictory.
There is no solution for such systems.
(E1)
(E2)
Example A.
Dependent Systems
Example B.
{x + y = 2
2x + 2y = 4
2*E1 – E2:
2x + 2y = 4
) 2x + 2y = 4
0 = 0
(E1)
(E2)
It’s a dependent system system.
There’re infinitely many solutions, e.g. (2, 0), (1, 1) etc..

2. EXERCISES 6.2
A. Given the augmented matrices for a system, classify the system as dependent, inconsistent or has
a unique solution.
1) 2) 3) 4)
B. By inspection, tell whether the system is dependent, inconsistent, or has a unique solution. Then
give the rref-form of the augmented matrix for the system.
5) x + y = -1 6) x + y = 2 7) 2x – 2y = 4 8) 2x + y = 0
x + y = 2 3x + 3y = 6 x – y = 2 2x + y = 3
9) x + y = 2 10) 2x + y = 9 11) x + y – 2z = 4 12) x + y + z = 3
x - y = 2 x – y = 3 x + z = 1 2x + 2y + 2z = 6
x + y – 2z = 0 x - z = 0
C. Solve by expressing the solutions in suitable variables. Give three different solutions for each
system.
13) Exercise #7 14) Exercise #6 15) Exercise #12
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3. (Answers to the odd problems) Exercise A.
1) Unique solution 3) Inconsistent
Exercise B.
5) Inconsistent 7) Dependent 9) Unique solution 11) Dependent
1 1 0 1 -1 2 1 0 2 1 0 1 0
0 0 1 0 0 0 0 1 0 0 1 -3 0
0 0 0 1
Exercise C.
13) y = x – 2 15) y = 3 – 2x, z = x