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System of linear equations
The task for solution
Solve the following system of equations
1.
3x – 7y + 5z = 4
2.
-2x + 5y - 3z = –1
5x – 3y + 4z = 11
3.
4x – 3y + 5z = -7
-3x + y - z =
7
5x – 4y + 6z = -10
5.
4x – 3y + 5z = 4
x - 2y + z = –2
5x – 5y + 6z = 3
7.
3x
2x
x
5x
9.
3x - 5y
x + 2y
-2x - 15y
–11y
+
+
–
–
8y
3y
3y
8y
+
+
–
+
3z
4z
2z
4z
4. 3x –
y + 5z = 4
6x - 2y + 10z = 8
9x – 3y + 15z = 12
6. x – 2y + 3z –
5u
2x + y + 4z +
u
3x – 3y + 8z – 2u
2x – 2y + 5z – 12u
– u = 4
+ u = –4
– 2u = 3
+ 2u = -8
+
z – u = 3
- 7z + u = 2
+ 36z - 6u = -7
+ 22z - 4u = -3
11.
-x
-x
-x
-2x
13.
3x
3x
-x
-x
15.
-x
-3x
-x
-2x
- 2y - 2z
+ y + 2z
+ y + z
– y + 2z
+
+
-
17.
2x
-3x
-x
2x
- y - 3z
+ 3y + 2z
- 3y – 3z
+ 3y + 2z
19.
-2x
-x
-3x
3x
21.
x + 2y + 5z – u + v = –4
2x
+ z + 2u – v = 8
3y + 2z – u + 2v = –1
x + y + z + u + v = 4
–x + 2y – z
+ v = 3
- 3y - 3z + 2u = -1
- 2y - 2z + 2u = 1
- y – 2z + 3u = -4
+ 2y + 2z + u = 14
+
+
2x – 3y + 5z = 7
-5x + 2y - 3z = –4
-x – 4y + 7z = 10
8. 3x – 7y +
= 2
= –3
= –1
= 4
5z + u =
x - y + 5z - u =
4x – 8y + 10z
=
5x – 9y + 15z – u =
10.
12.
3x
x
5x
-x
4
2
6
8
+ 2y - 4z + u =
5
- 3y + z - 2u =
2
+ y – 3z – 5u = -3
– 3y + 5z - 7u = -13
-3x
3x
x
2x
- 3y + 2z - u = 49
- y + z + 3u = -16
- 2y – 3z + u = -7
– y - z - 3u = 29
14.
2x
-2x
-x
-2x
- 2y + 3z
+ 3y + 3z
- 2y – z
+ y + 2z
- u = -14
+ 2u = 30
+ 2u = -18
- u =
8
= 18
= -17
=
3
=
3
16.
-x
3x
-2x
x
+ y +
- y - 2y –
+ y -
- u = -12
+ 2u = 26
– u = -6
- u = 12
- u = -2
- 3u = -27
+ 3u = 39
- u = -34
18.
3x
3x
x
-x
- 3y + 3z + 2u = -46
+ 3y - 3z + 3u =
5
+ 2y – 3z + u = -8
+ y - 3z - u = 35
20.
-3x
-3x
x
3x
3y
2y
3y
3y
- z + u
+ 3z + 2u
– 3z + u
+ 2z + 2u
2u
2u
2u
2u
= -7
= 40
= -21
= 35
2z
2z
2z
2z
- 2y - 3z + u = 52
- 2y + z - 3u = 12
+ 2y – 2z + u = 0
– y - z - u = 14
+
–
2y
3y
2y
2y
- 2z + u
- 2z + u
+ z + u
+ 2z - 2u
= 21
= 21
= -4
= -22
22.
x + 2y + 5z – u + v = –4
2x
+ z + 2u – v = 8
2y + 2z – u + 2v = –1
x + y + z + u + v = 4
–x + 2y – z
+ v = 3
23.
x - 2y +
3x – 4y +
-2x + y 2y -3y +
24.
3x
-5x
x
7x
–2x
25.
x
2x
3x
4x
26.
3x
-5x
x
7x
–2x
2x
27.
–4x1
4x1
–3x1
–x1
4x1
5x1
+
+
+
+
+
–
28.
–4x1
4x1
–3x1
–2x1
4x1
3x1
–x1
3x1
+
+
+
+
+
–
+
+
29.
–4x1
4x1
–3x1
–x1
4x1
x1
–2x1
3x1
x1
+
+
+
+
+
–
+
+
+
+
+
-
4y
y
3y
4y
3y
3z
5z
z
4z
5z
– 2u - 4v
- u + 2v
+ 3u - 5v
+ 5u - 4v
- u - 13v
+ 5z –
- 3z +
+
+ 5z + 2z +
u
4u
2u
3u
3u
= 5
= 4
= –2
= 3
= 8
+ 2v = 7
– 2v = 5
- 5v = –2
+ v = 7
= 10
- y + z + u - 2v = 0
+ y - z - u + v = 1
+ 3y - 3z – 3u + 4v = 2
+ 5y - 5z - 5u + 7v = 3
+
+
+
4y
y
3y
4y
3y
y
+ 5z –
- 3z +
+
+ z + 2z +
– 3z –
x2
2x2
3x2
2x2
2x2
2x2
5x2
3x1 + x2
u
4u
2u
3u
3u
2u
+ 2v + 4w
– 2v - w
- 5v
+ v + 2w
- 3w
+ v – w
= -17
= 19
= 12
= -16
= -16
=
3
–
+
–
–
–
–
–
+
5x3
5x3
x3
x3
4x3
2x3
3x3
x3
+ x4
+ x4
+ x4
+ x4
– 3x4
+ x4
+ 5x4
+ x4
–
+
+
–
–
+
+
+
x5
2x5
5x5
4x5
x5
4x5
3x5
x5
–
+
–
–
–
–
+
3x6
2x6
x6
x6
x6
3x6
5x6
2x6
+
+
–
+
–
+
+
+
x7
5x7
4x7
x7
x7
4x7
4x7
3x7
= 18
= 8
= 0
= 5
= 3
= 2
= 18
= 11
3x2
2x2
3x2
2x2
2x2
2x2
3x2
2x2
–
+
–
–
–
–
–
+
3x3
3x3
x3
x3
4x3
2x3
3x3
2x3
+
+
+
+
–
+
+
+
x4
2x4
x4
2x4
3x4
x4
3x4
x4
–
+
+
–
–
+
+
+
x5
2x5
3x5
4x5
x5
4x5
3x5
x5
–
+
–
–
–
–
-
3x6
2x6
x6
x6
2x6
3x6
3x6
2x6
+
+
–
+
–
+
+
+
x7
3x7
4x7
x7
2x7
4x7
4x7
3x7
–
+
+
–
–
–
+
–
3x8
2x8
2x8
2x8
x8
3x8
2x8
4x8
=
3
= 11
=
7
=
0
= -4
= -11
=
5
=
0
x2
x2
3x2
2x2
2x2
2x2
x2
x2
x2
– x3
+ x3
– 3x3
– 3x3
– 4x3
+
+
–
+
+
+
+
x4
x4
3x4
x4
3x4
x4
x4
2x4
2x4
–
+
+
–
–
+
+
+
+
2x5
x5
x5
4x5
2x5
4x5
3x5
3x5
3x5
–
+
–
–
–
–
+
3x6
2x6
3x6
3x6
x6
3x6
x6
+
x6
+
+
–
+
–
–
+
+
+
3x7
x7
4x7
2x7
x7
4x7
4x7
3x7
3x7
–
+
+
–
–
–
+
+
+
3x8
x8
2x8
x8
2x8
x8
x8
4x8
4x8
+ x9 = -9
- 2x9 = 6
+ x9 = -4
- 3x9 = -9
+ x9 = 1
- x9 = -6
+ 4x9 = -7
- x9 = -4
- x9 = -2
– 3x3
+ x3
+ x3
30.
–4x1
-4x1
–3x1
–2x1
4x1
x1
–x1
3x1
4x1
2x1
+
+
+
+
+
–
+
x2
x2
3x2
2x2
2x2
2x2
x2
– x3 + x4
+ 2x3 + x4
+ x4
+ x4
+ 4x3 – 3x4
+ 2x3 + x4
– 3x3
+ 2x4
+ 2x2 + 4x3 – 3x4
– 2x2 + 2x3 + x4
– x5 –
+ 2x5 +
+ x5 –
– 4x5
–
+ 4x5 –
+ 3x5
+
–
+ 4x5 –
3x6 +
2x6 + 5x7 +
x6 – 4x7 +
+ 8x7
3x6 – 6x7 –
3x6 – 4x7 –
+ 4x7 +
2x6 + 3x7 –
3x6 – 6x7 –
3x6 – 4x7 –
3x8 + 2x9 – x10
x8 + x9
2x8 + x9 – 2x10
+ 2x9 – x10
x8 - x9 – 2x10
5x8 + 3x9 – 3x10
x8 - x9 – x10
4x8 - x9 – 4x10
x8
– x10
5x8 + x9 – 2x10
= 13
= -18
=
5
= -5
= -3
=
9
= 11
= -3
= -7
=
5
Results:
1. System of the equations has only one solution: x = 1; y = 2; z = 3.
2. System of the equations has infinitely many solutions in R: x =
where t ∈ R .
1
2
19
27
t − ; y = t − ; z = t,
11 11
11 11
3. System of the equations has only one solution: x = −2; y = 3; z = 2.
4. System of the equations has infinitely many solutions in R: x =
where p, t ∈ R .
4 p 5
+ − t ; y = p; z = t ,
3 3 3
5. System of the equations does not have solution in R.
6. System of the equations has infinitely many solutions in R:
x = 30 + 71t ; y = −7 − 15t ; z = −14 − 32t ; u = t , where t ∈ R .
7. System of the equations has only one solution: x = 2; y = 1; z = −3; u = 1.
8. System of the equations has infinitely many solutions in R:
5 15
1 5
x= −
p + 2t ; y = − p + t ; z = p; u = t , where p, t ∈ R .
2 2
2 2
9. System of the equations has infinitely many solutions in R:
16
3
3
4
x=
+ 3 p − t ; y = + 2 p − t ; z = p; u = t , where p, t ∈ R .
11
11
11
11
10. System of the equations has infinitely many solutions in R:
74 9
83 7
127 19
x=−
+ t; y = −
+ t; z = −
+ t ; u = t , where t ∈ R .
24 4
24 8
24
8
11. System of the equations has only one solution: x = −5; y = −5; z = 7; u = 0.
12. System of the equations has only one solution: x = −1; y = −9; z = 5; u = −9.
13. System of the equations has only one solution: x = 1; y = 4; z = 5; u = 7.
14. System of the equations has only one solution: x = 1; y = 9; z = 1; u = 1.
15. System of the equations has only one solution: x = 2; y = −9; z = 4; u = 5.
16. System of the equations has only one solution: x = 8; y = −2; z = −2; u = −2.
17. System of the equations has only one solution: x = −3; y = −8; z = 0; u = 4.
18. System of the equations has only one solution: x = 2; y = −10; z = −8; u = 2.
19. System of the equations has only one solution: x = 7; y = 7; z = −1; u = −4.
20. System of the equations has only one solution: x = −10; y = 0; z = 5; u = 1.
21. System of the equations has only one solution: x = 1; y = 2; z = −1; u = 3; v = −1.
22. System of the equations has only one solution: x = 6,6; y = 3,6; z = −3,6; u = −1,4; v = −1,2.
23. System of the equations has infinitely many solutions in R:
7
73
21 9
13
41
7
x = t + ; y = t + ; z = t + , u = t ; v = − , where t ∈ R .
2
18
2
2
2
18
9
24. System of the equations does not have solution in R.
25. System of the equations has infinitely many solutions in R:
1 1
1 5
x = − + t ; y = − t + p + q; z = q, u = p; v = t , where q, v, t ∈ R .
3 3
3 3
26. System of the equations has only one solution: x = −1; y = 2; z = −3; u = 1; v = −1; w = 3.
27. System of the equations has only one solution:
x1 = −2; x 2 = 2; x3 = −3; x 4 = −1; x5 = 1; x6 = 3; x7 = 4.
28. System of the equations has only one solution:
x1 = −1; x 2 = 3; x3 = 0; x 4 = −1; x5 = 1; x6 = 3; x7 = 1; x8 = 0.
29. System of the equations has only one solution:
x1 = 0; x 2 = 1; x3 = 3; x 4 = −5; x5 = 2; x6 = 2; x7 = 1; x8 = −2; x9 = −1.
30. System of the equations has only one solution:
x1 = −3; x 2 = −2; x3 = −5; x 4 = −4; x5 = −1; x6 = −2; x7 = −1; x8 = −2; x9 = −1; x10 = −3.
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