Systems of Linear Equations
Table of Contents
Systems of Linear Equations ...............................................................................................2
Systems of Linear Equations ..................................................................................................2
SLE with Parameter ................................................................................................................6
SLE over Zp .............................................................................................................................9
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Systems of Linear Equations
Systems of Linear Equations
Systems of Linear Equations
Questions
For the following augmented matrix perform the indicated elementary
 3 −2 1 
row operations: 

 2 0 −1
2R1 → R1
R1  R2
R2 + R1 → R2
R2 + 2R1 → R2
For the following augmented matrix perform the indicated elementary
1 −2 0 3 
row operations:  4 0 −1 2 
 3 1 2 −4 
2R3 → R3
R1  R3
R3 − R1 → R3
R2 − 2 R1 → R2
For the following augmented matrix perform the indicated elementary
1 2 4 0
row operations:  −2 0 −1 2  .
 −1 2 2 −1
1
R3 → R3
2
R2  R3
R2 − 4 R1 → R2
R3 + 2 R1 → R3
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2
Systems of Linear Equations
For the following systems of equations, convert the system into an augmented matrix and use
the augmented matrix techniques to determine the solution to the system, or to determine if
the system is inconsistent:
2 x + 7 y = 13

2 x + 5 y = 11
2 x + 3 y = 7

4 x − 5 y = 3
2 x + 3 y = 8

5 x − 4 y = −3
4 x + 8 y = 20

 3 x + 6 y = 15
−6 x + 3 y = 15

 10 x − 5 y = −25
8 x − 4 y = 10

−6 x + 3 y = 1
 x + 2 y + 3 z = −11

 2 x + 3 y − z = −5
3 x + y − z = 2

2 x − y − 3z = 5

3x − 2 y + 2 z = 5
10 x − 6 y − 2 z = 32

 x + 2 y + 3z = 3

4 x + 6 y + 16 z = 8
3 x + 2 y + 17 z = 1

x + 3y = 2

2 x + y = −1
 x − y = −2

4 x − 7 y = 0

8 x − 14 y = 2
−16 x + 28 y = 0

3 x − 2 y = 1

 −9 x + 6 y = − 3
6 x − 4 y = 2

x + 2 y + 2z = 2
3 x − 2 y − z = 5


2 x − 5 y + 3 z = −4
2 x + 8 y + 12 z = 0
Solve the following SLE:
 z1 + iz2 + (1 − i ) z3 = 1 + 4i

a. iz + z2 + (1 + i ) z3 = 2 + i

( −1 + 3i ) z1 + ( 3 − i ) z2 + ( 2 + 4i ) z3 = 5 − i
Over the field (Complex numbers).
 z1 + iz2 + (1 − i ) z3 = 1 + 4i

b. iz + z2 + (1 + i ) z3 = 2 + i

( −1 + 3i ) z1 + ( 3 − i ) z2 + ( 2 + 4i ) z3 = 5 − i
Over the field
(Real numbers).
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Systems of Linear Equations
Answer Key
 3 −2 1 
6 −4 2 
 3 −2 1 
 2 0 −1
a. 
b. 
→ 
→ 




 2 0 −1 2 R1→R1  2 0 −1
 2 0 −1 R1  R2  3 −2 1 
 3 −2 1 
3 −2 1 
 3 −2 1 
3 −2 1
c. 
d. 
→ 
→ 




 2 0 −1 R2 + R1→ R2 5 −2 0 
 2 0 −1 R2 + 2 R1→R2 8 −4 1
1 −2 0 3 
1 −2 0 3 
 4 0 −1 2 
→
a.  4 0 −1 2 


2 R3 → R3
 3 1 2 −4 
 6 2 4 −8
1 −2 0 3 
b.  4 0 −1 2 
 3 1 2 −4 
1 −2 0 3 
c.  4 0 −1 2 
 3 1 2 −4 
1 −2 0 3 
d.  4 0 −1 2 
 3 1 2 −4 
1 2 4 0
a.  −2 0 −1 2 
 −1 2 2 −1
1
b.  −2
 −1
1
c.  −2
 −1
2 4 0
0 −1 2 
2 2 −1
2 4 0
0 −1 2 
2 2 −1
1 2 4 0
d.  −2 0 −1 2 
 −1 2 2 −1
→
R1  R3
→
R3 − R1 → R3
→
R2 −2 R1 → R2
→
1
R → R3
2 3
→
R2  R3
3
4

1
1
4

 2
1 2 −4 
0 −1 2 
−2 0 3 
−2 0 3 
0 −1 2 
3 2 −7 
1 −2 0 3 
 2 4 −1 −4 


 3 1 2 −4 
2 4
0 
 1
 −2 0 −1 2 


 −0.5 1 1 −0.5
1 2 4 0
 −1 2 2 −1


 −2 0 −1 2 
→
4
0
1 2
 −6 −8 −17 2 


 −1 2
2 −1
→
1 2 4 0
 −2 0 −1 2 


 1 6 10 −1
R2 − 4 R1 → R2
R3 + 2 R1 → R3
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Systems of Linear Equations
( 3,1)
( 2,1)
(1, 2 )
( 5 − 2t , t ) ,  solutions.
( −2.5 + 0.5t , t ) ,  solutions.
No solution.
(1, −3, −2 )
No solution.
 solutions.
( −1,1)
No solution.
 solutions.
( 2,1, −1)
a. z1 = 1 + i − (1 − i ) t , z 2 = 3, z3 = t
b. z1 = 2, z2 = 3, z3 = −1
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5
Systems of Linear Equations
SLE with Parameter
Questions
Determine the values of k for which the system below has no solutions, exactly one solution,
or infinitely many solutions:
x − y + z = 1

2
2
5 x − 7 y + k + 3 z = k + 1

3 x − y + ( k + 3) z = 3
 x + ky + z = 1

 x + y + kz = 1
kx + y + z = 1

 x + 2ky + z = 0

3 x + y + kz = 2
 x + 9ky + 5 z = −2

2 x − y + z = 0

x + 2 y − z = 0
5 x + 1 − k y + k 2 z = 1
( )


kx − y =1

( k − 2 ) x + ky = −2

2
 ( k − 1) z = 9
2 x + ky = 3

2
( k + 3) x + 2 y = k + 5

2
6 x + 3ky = 7k + 2
3 x + 4 y − z = 2
kx − 2 y + z = −1


 x + 8 y − 3z = k
2 x + 6 y − 2 z = 0.5k + 1
 x + ky + 3z = 2

kx − y + z = 4
3x + y + 2 + k z = 0
(
)

(
)
2 x − 3 y + z = 1

2
4 x + k − 5k y + 2 z = k
(
)
Determine the values of a and b , for which the systems below has no solutions,
exactly one solution, or infinitely many solutions:
2 x + 4 y + az = −1
 x + 2 y + 4 z = −4


 x + 2 y − 4z = 0
 x + 2 y + 6 z = −2b
x + y − z + t = 1

ax + y + z + t = b
3 x + 2 y + at = 1 + a

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Systems of Linear Equations
Determine the following:
a. Relationship between the values of a , b , c and d for which the system below has
exactly one solution.
b. The values of b , c and d for which the system below has infinitely many solutions for all
values of a .
 x + az = 1

 y + 2z = 2
bx + cy + dz = 3

 x + y − z =1

Given the SLE  3 x − 7 y + k 2 + 1 z = k 2 − 1 .

4 x − 6 y + ( k + 2 ) z = 4
a. Write the matrix corresponding to the system.
b. Bring the matrix to row-echelon form.
c. Find the values of k for which the SLE has [no, one, infinitely many solution(s)].
(
)
x + y − z = 1


Continuing with the SLE (in echelon-form):  − 10 y + k 2 + 4 z = k 2 − 4

2
2
 −k + k + 2 z = 4 − k
d. Write the general solution for the case when there are infinitely many solutions.
e. For which value of k does the SLE have a solution with z = 0 ?
f. For which value of k does the SLE have a unique solution with z = 0 ?
(
(
)
)
 x + y − z =1

Given the SLE  3 x − 7 y + k 2 + 1 z = k 2 − 1

4 x − 6 y + ( k + 2 ) z = 4
g. For which value of k will ( x, y, z ) = (1, 2,3) be a solution of the 3rd equation?
(
)
h. For which value of k is ( x, y, z ) = (1, 2,3) a solution of the SLE?
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Systems of Linear Equations
Answer Key
One solution: k  −2, k  1 ;
Infinitely many solutions: k = −2 ; No solutions: k = 1 .
One solution: k  1, k  −2 ;
Infinitely many solutions: k = 1 ;
No solutions: k = −2 .
4
4
One solution: k  −1, k  ;
Infinitely many solutions: k = −1 ; No solutions: k = .
7
7
One solution: k  1, k  −0.4 ; No solutions: k = 1 or k = −0.4 .
No solutions: k = 1 , k = −1 , k = −2 ;
Otherwise: Single solutions.
One solution: k = −1 , k = 2 , k = −3 ; Otherwise: No solutons.
One solution: k  −1 ;
Infinitely many solutions: k = 1 ;
No solutions: k  1 .
Infinitely many solutions: k  3 ; No solutions: k = 3 .
One solution: k  1 ;
No solutions: k = 1 .
No solutions: 5 − 2b  0 or 3 + 0.5a  0 , b  2.5 or a  −6 ;
Infinitely many solutions: b = 2.5 and a = −6 .
No solutions: a = 2, b  2 ;
Infinitely many solutions: a  2 or a = b = 2 .
a. One solution: d − ab − 2c  0 ; b. Infinitely many solutions for all a : b = 0, c = 1.5, d = 3 .
−1
1 
1 1

2
2
a.  3 −7 k + 1 k − 1
 4 −6 k + 2
4 
c. One solution: ,k  2, k  −1 ;
d. x = 1 + 0.2t , y = 0.8t , z = t
g. k = 2
−1
1 
1 1

2
2
k +4
k − 4 
b. 0 −10
0 0 −k 2 + k + 2 4 − k 2 
Infinitely many solutions: k = 2 ;
No solutions: k = −1 .
e. x = 1, y = 0, z = 0
f. k = −2
h. No value of k .
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8
Systems of Linear Equations
SLE over Zp
Questions
Solve the following:
2 x − y = 3
a. System of linear equations 
over the field
x + 2 y = 4
b. Same SLE over 5 = 0,1, 2,3, 4 .
 x + 2 y + 3z = 1

Solve the SLE 2 x + 4 y + 4 z = 2 over the field
3 x + z = 0

3 x + y + 4 z = 3

Solve the SLE 4 x + 3 y + 3 z = 4 over the field
2 x + 4 z = 0

5
= 0,1, 2,3, 4 .
5
= 0,1, 2,3, 4 .
 x + 4 y + 2 z + 4t = 1
x + 2 y − z = 0

Solve the SLE 
over the field
y
+
z
+
t
=
1

 x + 3 y − z − 2t = 0
 x + ky + z = 1

Given the system  x + y + kz = 1 over the field
kx + y + z = 1

.
7
3
= 0,1, 2,3, 4,5, 6 .
= 0,1, 2 .
Determine the values of k for which the system has: exactly one solution; no solutions;
infinitely many solutions; other.
x − y + z = 1

Given the system 3 y + k 2 + 3 z = k 2 + 1 over 5 = 0,1, 2,3, 4 .

3 x − y + ( k + 3) z = 3
Determine the values of k , for which the system has:
No solutions.
Exactly one solution.
Infinitely many solutions.
Other.
(
a.
b.
c.
d.
)
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9
Systems of Linear Equations
Answer Key
a. ( 2,1)
b. x = 4 + 3t , y = t; Just 5 solutions (not  ) for t = 0,1, 2,3, 4 : ( 4, 0 )( 2,1)( 0, 2 )( 3,3)(1, 4 ) .
( 0,3, 0 )
( 0,3, 0 )
x = 1, y = 4, z = 2, t = 2
One solution: k = 0, 2 , There are no Infiniate solutions and No solutions.
a. No solutions: k = 1 .
b. One solution: k  3, k  1 .
c. There are no infinite solutions.
d. Five Solutions: k = 3 .
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10