Sistemi di equazioni di I grado letterali con discussione

Sistemi di equazioni di I grado letterali con discussione
Algebra
1
2
�
�
(𝑎𝑎 − 4)𝑥𝑥 − 2𝑦𝑦 = 3
(𝑎𝑎 − 3)𝑥𝑥 − 𝑦𝑦 = 𝑎𝑎 − 2
(𝑚𝑚 − 2)𝑥𝑥 + 3𝑦𝑦 = 6
�
(𝑚𝑚 − 1)𝑥𝑥 + 4𝑦𝑦 = 8
4
�𝑎𝑎2
6
7
8
9
10
11
v 3.0
�
1
�
(0; 2)
1
C. E. 𝑎𝑎 ≠ 0, 𝑏𝑏 ≠ 0
𝑥𝑥 − 𝑏𝑏 𝑦𝑦 = 0
2𝑎𝑎
2𝑏𝑏
� 2
; 2
�
𝑏𝑏 + 4 𝑏𝑏 + 4
𝑏𝑏
C. E. 𝑎𝑎 ≠ 0, 𝑏𝑏 ≠ 0
se 𝑏𝑏 = ±1 impossibile
se 𝑏𝑏 ≠ ±1 determinato
1
1
𝑥𝑥 + 𝑦𝑦 = 0
𝑏𝑏
�𝑎𝑎
1
𝑥𝑥 + 𝑏𝑏𝑏𝑏 = 1
𝑎𝑎
𝑎𝑎
𝑏𝑏
�
;
�
1 − 𝑏𝑏 2 𝑏𝑏 2 − 1
se 𝑎𝑎 = 0 impossibile
se 𝑏𝑏 ≠ 0 impossibile
se 𝑏𝑏 = 0 indeterminato
𝑏𝑏𝑏𝑏 + 𝑎𝑎𝑎𝑎 = 3
� 2
𝑏𝑏 𝑥𝑥 + 𝑎𝑎𝑎𝑎𝑎𝑎 = 2𝑏𝑏
1
1
𝑥𝑥 − 𝑦𝑦 = 0
𝑏𝑏
�𝑎𝑎
1
2
𝑥𝑥 − 𝑦𝑦 = 1
𝑎𝑎
𝑏𝑏
1
1
𝑥𝑥 − 𝑦𝑦 = 5
𝑏𝑏
�𝑎𝑎
1
2
𝑥𝑥 − 𝑦𝑦 = 0
𝑎𝑎
𝑏𝑏
�
3(𝑚𝑚 + 2) 2𝑚𝑚 − 11
;
�
2(𝑚𝑚 − 1) 2(𝑚𝑚 − 1)
se 𝑚𝑚 = 5 indeterminato
se 𝑚𝑚 ≠ 5 determinato
𝑥𝑥 + 2 𝑦𝑦 = 1
𝑎𝑎
�
2𝑎𝑎 − 7 𝑎𝑎2 − 9𝑎𝑎 + 17
;
�
𝑎𝑎 − 2
𝑎𝑎 − 2
se 𝑚𝑚 = 1 impossibile
se 𝑚𝑚 ≠ 1 determinato
𝑥𝑥 + 𝑚𝑚𝑚𝑚 = 3 − 𝑚𝑚
2𝑥𝑥 + 2𝑦𝑦 = −5
3
5
se 𝑎𝑎 = 2 impossibile
se 𝑎𝑎 ≠ 2 determinato
C. E. 𝑎𝑎 ≠ 0, 𝑏𝑏 ≠ 0
(−𝑎𝑎; −𝑏𝑏)
C. E. 𝑎𝑎 ≠ 0, 𝑏𝑏 ≠ 0
(10𝑎𝑎; 5𝑏𝑏)
3𝑥𝑥 − 6𝑦𝑦 = 𝑚𝑚
2𝑥𝑥 − 4𝑦𝑦 = 𝑚𝑚 + 1
se 𝑚𝑚 = −3 indeterminato
se 𝑚𝑚 ≠ −3 𝑖𝑖mpossibile
se 𝑚𝑚 = 4, 𝑛𝑛 = 1/3 indeterminato
se 𝑚𝑚 = 4, 𝑛𝑛 ≠ 1/3 impossibile
se 𝑚𝑚 ≠ 4 determinato
6𝑛𝑛 + 6 − 2𝑚𝑚 1 − 3𝑛𝑛
�
;
�
12 − 3𝑚𝑚
12 − 3𝑚𝑚
2𝑥𝑥 + 𝑚𝑚𝑚𝑚 = 𝑛𝑛 + 1
3𝑥𝑥 + 6𝑦𝑦 = 2
se 𝑎𝑎 ≠ 0, impossibile
se 𝑎𝑎 = 5𝑏𝑏/3 con 𝑏𝑏 = 0
indeterminato
𝑎𝑎𝑎𝑎 + 𝑦𝑦 = 3
� 2
𝑎𝑎 𝑥𝑥 + 𝑎𝑎𝑎𝑎 = 5𝑏𝑏
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1 di 7
Algebra
Sistemi di equazioni di I grado letterali con discussione
12
(𝑚𝑚 − 2)𝑥𝑥 − 3𝑦𝑦 = 3
�
(2𝑚𝑚 + 5)𝑥𝑥 − 2𝑦𝑦 = −1
13
(𝑎𝑎 + 2)𝑥𝑥 + 3𝑦𝑦 = 1
�
(𝑎𝑎 − 3)𝑥𝑥 + 2𝑦𝑦 = 5
14
15
�
�
se 𝑚𝑚 = −19/4 impossibile
se 𝑚𝑚 ≠ −19/4 determinato
9
13 + 7𝑚𝑚
�−
;−
�
19 + 4𝑚𝑚
19 + 4𝑚𝑚
se 𝑎𝑎 = 13 impossibile
se 𝑎𝑎 ≠ 13 determinato
13
4𝑎𝑎 + 13
�
;
�
𝑎𝑎 − 13 13 − 𝑎𝑎
𝑥𝑥 + 𝑦𝑦 = 𝑚𝑚 + 1
𝑥𝑥 − 𝑦𝑦 = 1 − 𝑚𝑚
sistema determinato ∀ 𝑚𝑚
(1; 𝑚𝑚)
se 𝑚𝑚 = 0 𝑛𝑛 = 0 indeterminato
se 𝑚𝑚 = 0 𝑛𝑛 ≠ 0 impossibile
se 𝑚𝑚 ≠ 0 determinato
𝑥𝑥 + 𝑚𝑚𝑚𝑚 = 𝑚𝑚 − 1
𝑥𝑥 − 𝑚𝑚𝑚𝑚 = 𝑛𝑛2 − 1
�
𝑚𝑚 + 𝑛𝑛2 − 2 𝑚𝑚 − 𝑛𝑛2
;
�
2
2𝑚𝑚
1
16
17
18
19
v 3.0
1
�
�
6
se 𝑎𝑎 = 2𝑏𝑏−1 𝑏𝑏 = 11 indeterminato
(𝑎𝑎 + 1)𝑥𝑥 + 𝑏𝑏𝑏𝑏 = 6
�
(𝑎𝑎 − 1)𝑥𝑥 − (𝑏𝑏 − 1)𝑦𝑦 = 5
�
6
se 𝑎𝑎 = 2𝑏𝑏−1 𝑏𝑏 ≠ 11 impossibile
1
se 𝑎𝑎 ≠ 2𝑏𝑏−1 determinato
6 − 11𝑏𝑏
11 − 𝑎𝑎
�
;
�
𝑎𝑎 − 2𝑎𝑎𝑎𝑎 + 1 𝑎𝑎 − 2𝑎𝑎𝑎𝑎 + 1
se 𝑏𝑏 = −4 𝑎𝑎 = 1 indeterminato
se 𝑏𝑏 = −4 𝑎𝑎 ≠ 1 impossibile
se 𝑏𝑏 ≠ −4 determinato
𝑥𝑥 − (𝑏𝑏 + 2)𝑦𝑦 = 2
2𝑥𝑥 + 4𝑦𝑦 = 4𝑎𝑎
4 + 2𝑎𝑎𝑎𝑎 + 4𝑎𝑎 2𝑎𝑎 − 2
;
�
𝑏𝑏 + 4
𝑏𝑏 + 4
�
se 𝑎𝑎 = −
se 𝑎𝑎 = −
(𝑎𝑎 + 3)𝑥𝑥 − 5𝑏𝑏𝑏𝑏 = 6
2𝑥𝑥 + 4𝑦𝑦 = 1
se 𝑎𝑎 ≠ −
5𝑏𝑏+6
2
5𝑏𝑏+6
2
5𝑏𝑏+6
2
𝑏𝑏 = −
𝑏𝑏 ≠ −
24
5
24
5
indeterminato
impossibile
determinato
24 + 5𝑏𝑏
𝑎𝑎 − 9
�
;
�
4𝑎𝑎 + 12 + 10𝑏𝑏 4𝑎𝑎 + 12 + 10𝑏𝑏
se 𝑘𝑘 = 2 impossibile
se 𝑘𝑘 ≠ 2 determinato
4𝑥𝑥 + 𝑦𝑦 = 𝑘𝑘
(10 − 𝑘𝑘)𝑥𝑥 + 2𝑦𝑦 = 10
�
© 2016 - www.matematika.it
2𝑘𝑘 − 10 𝑘𝑘 2 − 10𝑘𝑘 + 40
;
�
𝑘𝑘 − 2
𝑘𝑘 − 2
2 di 7
Algebra
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�
Sistemi di equazioni di I grado letterali con discussione
(𝑎𝑎 + 1)𝑥𝑥 + 𝑎𝑎𝑎𝑎 = 5
(𝑎𝑎2 − 1)𝑥𝑥 + 𝑎𝑎(𝑎𝑎 − 1)𝑦𝑦 = 3𝑎𝑎 − 5
se 𝑎𝑎 = 0 indeterminato
se 𝑎𝑎 ≠ 0 impossibile
se 𝑎𝑎 = 6/7 impossibile
se 𝑎𝑎 ≠ 6/7 determinato
3𝑎𝑎𝑎𝑎 + (𝑎𝑎 + 3)𝑦𝑦 = 1
�
(3𝑎𝑎 − 2)𝑥𝑥 + 𝑎𝑎𝑎𝑎 = 3
�
2𝑎𝑎 + 9 6𝑎𝑎 + 2
�
;
�
7𝑎𝑎 − 6 6 − 7𝑎𝑎
𝑥𝑥 + 𝑦𝑦 = 𝑎𝑎 + 𝑏𝑏
𝑎𝑎𝑎𝑎 + 𝑏𝑏(𝑦𝑦 − 2𝑏𝑏) = (𝑎𝑎 − 𝑏𝑏)2
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ 𝑏𝑏: 𝑥𝑥 = 𝑎𝑎 − 2𝑏𝑏, 𝑦𝑦 = 3𝑏𝑏
𝑠𝑠𝑠𝑠 𝑎𝑎 = 𝑏𝑏: 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
23
𝑥𝑥 + 𝑦𝑦 = 𝑎𝑎 − 1
� 𝑎𝑎𝑎𝑎 + 𝑦𝑦 = 0
(𝑎𝑎 − 1)𝑥𝑥 − 𝑦𝑦 = 3
𝑠𝑠𝑠𝑠 𝑎𝑎 = −1: 𝑥𝑥 = −1, 𝑦𝑦 = −1
𝑠𝑠𝑠𝑠 𝑎𝑎 = 1: 𝑥𝑥 = 3, 𝑦𝑦 = −3
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ ±1: 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
24
𝑎𝑎𝑎𝑎 − 𝑦𝑦 = −1
� 𝑦𝑦 = 𝑎𝑎 + 1
2𝑎𝑎𝑎𝑎 − 𝑦𝑦 = 𝑎𝑎 − 1
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ 0: 𝑥𝑥 = 1, 𝑦𝑦 = 𝑎𝑎 + 1
𝑠𝑠𝑠𝑠 𝑎𝑎 = 0: 𝑥𝑥 = 𝑥𝑥, 𝑦𝑦 = 1
25
26
27
�
2𝑥𝑥 + 𝑦𝑦 = 𝑎𝑎 + 2
𝑎𝑎𝑎𝑎 + (𝑎𝑎 − 1)𝑦𝑦 = 2𝑎𝑎
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ 2: 𝑥𝑥 = 𝑎𝑎 + 1, 𝑦𝑦 = −𝑎𝑎
𝑠𝑠𝑠𝑠 𝑎𝑎 = 2: 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ 0, 𝑎𝑎 ≠ ±2:
𝑎𝑎2 + 2𝑎𝑎 − 4
4
𝑥𝑥 =
, 𝑦𝑦 =
𝑎𝑎(𝑎𝑎 + 2)
4 − 𝑎𝑎2
𝑠𝑠𝑠𝑠 𝑎𝑎 = 0, 𝑎𝑎 = ±2: 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝑎𝑎(𝑥𝑥 − 𝑦𝑦) + 2𝑦𝑦 = 𝑎𝑎
�
𝑎𝑎(𝑎𝑎𝑎𝑎 + 2𝑥𝑥 − 2) = (𝑎𝑎 + 2)(𝑎𝑎 − 2)
�
(𝑎𝑎 − 1)𝑥𝑥 + (𝑎𝑎 + 1)𝑦𝑦 = −2𝑎𝑎
(𝑎𝑎 + 1)𝑥𝑥 + (𝑎𝑎 − 1)𝑦𝑦 = 2𝑎𝑎
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ 0: 𝑥𝑥 = 𝑎𝑎, 𝑦𝑦 = −𝑎𝑎
𝑠𝑠𝑠𝑠 𝑎𝑎 = 0: 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
28
𝑥𝑥 + 𝑦𝑦 = 𝑎𝑎
𝑥𝑥
� + 𝑎𝑎𝑎𝑎 = 0
𝑥𝑥 − 𝑎𝑎𝑎𝑎 = 1
𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
29
𝑥𝑥 + 𝑎𝑎𝑎𝑎 = 1
� 𝑥𝑥 − 𝑦𝑦 = 1
𝑎𝑎𝑎𝑎 − 𝑦𝑦 = 0
𝑠𝑠𝑠𝑠 𝑎𝑎 = 0: 𝑥𝑥 = 1, 𝑦𝑦 = 0
1
1
𝑠𝑠𝑠𝑠 𝑎𝑎 = −1: 𝑥𝑥 = ,
𝑦𝑦 = −
2
2
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ −1, 𝑎𝑎 ≠ 0: 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
v 3.0
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3 di 7
Algebra
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v 3.0
Sistemi di equazioni di I grado letterali con discussione
2𝑥𝑥 − 𝑦𝑦 = 𝑎𝑎
𝑥𝑥
� + 𝑎𝑎𝑎𝑎 = 1
𝑥𝑥 + (𝑎𝑎 + 1)𝑦𝑦 = 3
𝑠𝑠𝑠𝑠 𝑎𝑎 = 0: 𝑥𝑥 = 1, 𝑦𝑦 = 2
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ 0: 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
2𝑥𝑥 + 𝑦𝑦
2𝑎𝑎
− 2
=0
(𝑎𝑎 − 1)
𝑎𝑎
�
2𝑥𝑥 + 𝑦𝑦
𝑦𝑦 −
= 2(1 − 𝑥𝑥)
𝑎𝑎2
�
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ 0 , 𝑎𝑎 ≠ ±1: 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝑠𝑠𝑠𝑠 𝑎𝑎 = 0, 𝑏𝑏 = 0: 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝑏𝑏
1
𝑠𝑠𝑠𝑠 𝑏𝑏 ≠ 0, 𝑎𝑎 ≠ − : 𝑥𝑥 = ,
2
2
2𝑎𝑎 − 𝑏𝑏
𝑦𝑦 =
2(𝑎𝑎 − 𝑏𝑏)
𝑏𝑏
𝑠𝑠𝑠𝑠 𝑏𝑏 ≠ 0, 𝑎𝑎 = − : 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
2
1
𝑠𝑠𝑠𝑠 𝑏𝑏 = 0, 𝑎𝑎 ≠ 0: 𝑥𝑥 = ,
2
2𝑎𝑎 − 𝑏𝑏
𝑦𝑦 =
2(𝑎𝑎 − 𝑏𝑏)
(2𝑎𝑎 + 𝑏𝑏)𝑥𝑥 + (𝑎𝑎 − 𝑏𝑏)𝑦𝑦 = 2𝑎𝑎
𝑎𝑎(2𝑥𝑥 − 𝑦𝑦) + 𝑏𝑏(𝑥𝑥 + 𝑦𝑦) = 𝑏𝑏
𝑥𝑥 𝑦𝑦
+ =1
𝑎𝑎
𝑏𝑏
�𝑥𝑥 𝑦𝑦
− =2
𝑎𝑎 𝑏𝑏
𝑥𝑥 𝑦𝑦
5
+ =
� 𝑎𝑎 𝑏𝑏 𝑎𝑎𝑎𝑎
3𝑥𝑥 2𝑦𝑦
5
−
=−
𝑎𝑎
𝑏𝑏
𝑎𝑎𝑎𝑎
�
𝑎𝑎𝑎𝑎 + 2𝑦𝑦 = 0
𝑏𝑏𝑏𝑏 + 4𝑦𝑦 = 0
3
𝑏𝑏
𝑎𝑎, 𝑦𝑦 = −
2
2
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ 0, 𝑏𝑏 ≠ 0: 𝑥𝑥 =
1
4
, 𝑦𝑦 =
𝑏𝑏
𝑎𝑎
𝑠𝑠𝑠𝑠 𝑏𝑏 = 2𝑎𝑎: 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝑠𝑠𝑠𝑠 𝑏𝑏 ≠ 2𝑎𝑎: 𝑥𝑥 = 0, 𝑦𝑦 = 0
𝑥𝑥
𝑦𝑦
1
+
=
�2𝑎𝑎 + 𝑏𝑏 2𝑎𝑎 − 𝑏𝑏 2𝑎𝑎 − 𝑏𝑏
𝑥𝑥 − 1
𝑦𝑦
=
2𝑎𝑎 + 𝑏𝑏 2𝑎𝑎 − 𝑏𝑏
�
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ 0, 𝑏𝑏 ≠ 0: 𝑥𝑥 =
𝑠𝑠𝑠𝑠 𝑏𝑏 ≠ ±2𝑎𝑎: 𝑥𝑥 =
2𝑎𝑎
𝑏𝑏
, 𝑦𝑦 =
2𝑎𝑎 − 𝑏𝑏
2𝑎𝑎 + 𝑏𝑏
10
8
, 𝑏𝑏 = : 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
3
3
10
8
𝑠𝑠𝑠𝑠 𝑎𝑎 =
, 𝑏𝑏 ≠ : 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
3
3
𝑠𝑠𝑠𝑠 𝑎𝑎 =
𝑎𝑎𝑎𝑎 + 2𝑦𝑦 = 𝑏𝑏
5𝑥𝑥 + 3𝑦𝑦 = 4
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠
© 2016 - www.matematika.it
10
8 − 3𝑏𝑏
5𝑏𝑏 − 4𝑎𝑎
∶ 𝑥𝑥 =
, 𝑦𝑦 =
3
10 − 3𝑎𝑎
10 − 3𝑎𝑎
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Sistemi di equazioni di I grado letterali con discussione
𝑠𝑠𝑠𝑠 𝑎𝑎 = −
38
�
5𝑏𝑏
5
, 𝑎𝑎 = : 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
2
5𝑏𝑏
5
𝑠𝑠𝑠𝑠 𝑎𝑎 = − , 𝑎𝑎 ≠ : 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
3
2
𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏 = 2
5𝑥𝑥 − 3𝑦𝑦 = 4
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠
3
−5𝑏𝑏
16𝑎𝑎 − 10𝑏𝑏
∶ 𝑥𝑥 =
,
3
5(5𝑏𝑏 + 3𝑎𝑎)
10𝑏𝑏 − 4𝑎𝑎
𝑦𝑦 =
5𝑏𝑏 + 3𝑎𝑎
2
39
40
�
10−4𝑎𝑎
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ 0, 𝑏𝑏 = 0: 𝑥𝑥 = , 𝑦𝑦 =
𝑎𝑎
3𝑎𝑎
5
6 + 4𝑏𝑏
10 − 4𝑎𝑎
𝑠𝑠𝑠𝑠 𝑎𝑎 < , 𝑏𝑏 ≠ 0: 𝑥𝑥 =
, 𝑦𝑦 =
2
3𝑎𝑎 + 5𝑏𝑏
3𝑎𝑎 + 5𝑏𝑏
5
3𝑎𝑎
6 + 4𝑏𝑏
𝑠𝑠𝑠𝑠 𝑎𝑎 < , 𝑏𝑏 ≠ − : 𝑥𝑥 =
,
2
5
3𝑎𝑎 + 5𝑏𝑏
10 − 4𝑎𝑎
𝑦𝑦 =
3𝑎𝑎 + 5𝑏𝑏
4
10 − 4𝑎𝑎
5
𝑠𝑠𝑠𝑠 𝑎𝑎 = , 𝑏𝑏 ≠ 0: 𝑥𝑥 = , 𝑦𝑦 =
5
5𝑏𝑏
2
3
4
10 − 4𝑎𝑎
5
𝑠𝑠𝑠𝑠 𝑎𝑎 = , 𝑏𝑏 ≠ − : 𝑥𝑥 = , 𝑦𝑦 =
2
5
5𝑏𝑏
2
3
5
𝑠𝑠𝑠𝑠 𝑎𝑎 = , 𝑏𝑏 = − : 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
2
2
6 + 4𝑏𝑏
10 − 4𝑎𝑎
5
, 𝑦𝑦 =
𝑠𝑠𝑠𝑠 𝑎𝑎 > , 𝑏𝑏 ≠ 0: 𝑥𝑥 =
3𝑎𝑎 + 5𝑏𝑏
3𝑎𝑎 + 5𝑏𝑏
2
3
6 + 4𝑏𝑏
5
,
𝑠𝑠𝑠𝑠 𝑎𝑎 > , 𝑏𝑏 ≠ 𝑎𝑎: 𝑥𝑥 =
5
3𝑎𝑎 + 5𝑏𝑏
2
10 − 4𝑎𝑎
𝑦𝑦 =
3𝑎𝑎 + 5𝑏𝑏
𝑥𝑥 + 𝑎𝑎𝑎𝑎 = 𝑏𝑏
5𝑥𝑥 − 5𝑦𝑦 = 4
𝑥𝑥 𝑦𝑦 + 𝑎𝑎
�𝑎𝑎 − 𝑎𝑎 = −2
𝑥𝑥 − 𝑦𝑦 = −𝑎𝑎
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ 0: 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
41
�
𝑥𝑥 + 𝑦𝑦 = 𝑎𝑎
𝑏𝑏𝑏𝑏 + 𝑎𝑎𝑎𝑎 = 𝑏𝑏 2
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ 𝑏𝑏: 𝑥𝑥 = 𝑎𝑎 + 𝑏𝑏, 𝑦𝑦 = −𝑏𝑏;
𝑠𝑠𝑠𝑠 𝑎𝑎 = 𝑏𝑏 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖.
42
�
𝑥𝑥 + 𝑦𝑦 = 𝑏𝑏
𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏 = 𝑎𝑎2
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ 𝑏𝑏: 𝑥𝑥 = 𝑎𝑎 + 𝑏𝑏, 𝑦𝑦 = −𝑎𝑎 ;
𝑠𝑠𝑠𝑠 𝑎𝑎 = 𝑏𝑏 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝑎𝑎𝑎𝑎 + (𝑎𝑎 + 1)𝑦𝑦 = 2
(𝑎𝑎2 + 𝑎𝑎)(𝑥𝑥 − 𝑦𝑦) = 1
1
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ 0 ∧ 𝑎𝑎 ≠ −1 ∧ 𝑎𝑎 ≠ − , 𝑥𝑥
2
1
1
= , 𝑦𝑦 =
;
𝑎𝑎
𝑎𝑎 + 1
𝑠𝑠𝑠𝑠 𝑎𝑎 = 0 ∨ 𝑎𝑎 = −1 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖;
1
𝑠𝑠𝑠𝑠 𝑎𝑎 = −
𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
2
43
v 3.0
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Sistemi di equazioni di I grado letterali con discussione
(𝑎𝑎 + 𝑏𝑏)𝑥𝑥 − 𝑏𝑏𝑏𝑏 = 𝑏𝑏
𝑎𝑎𝑎𝑎 + 2𝑏𝑏𝑏𝑏 = 3𝑎𝑎𝑎𝑎
2
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ − 𝑏𝑏 ∧ 𝑏𝑏 ≠ 0:
3
= 𝑎𝑎;
2
𝑠𝑠𝑠𝑠 𝑎𝑎 = − 𝑏𝑏 ∨ 𝑏𝑏
3
= 0 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
2
𝑎𝑎 ≠ −2 ∧ 𝑎𝑎 ≠ 1
𝑠𝑠𝑠𝑠 𝑎𝑎 = 0 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
1
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ 0 𝑥𝑥 =
, 𝑦𝑦 = 𝑎𝑎 − 1
𝑎𝑎 + 2
(𝑎𝑎 + 2)𝑥𝑥 + 𝑦𝑦 = 𝑎𝑎
𝑦𝑦
𝑎𝑎
�
2𝑥𝑥 −
+
=0
𝑎𝑎 − 1 𝑎𝑎 + 2
�
�
𝑥𝑥 = 𝑏𝑏, 𝑦𝑦
𝑠𝑠𝑠𝑠 𝑏𝑏 = 0 ∨ 𝑏𝑏
= 1 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖;
𝑠𝑠𝑠𝑠 𝑏𝑏 ≠ 0 ∧ 𝑏𝑏 ≠ 1 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
(𝑏𝑏 + 1)𝑥𝑥 − (𝑏𝑏 2 − 1)𝑦𝑦 = 𝑏𝑏 2 + 1
𝑥𝑥 − (𝑏𝑏 − 1)𝑦𝑦 = 1
𝑎𝑎 + 5
𝑏𝑏, 𝑦𝑦
5
3𝑎𝑎 − 10
=
;
5𝑎𝑎
𝑠𝑠𝑠𝑠 𝑎𝑎 = 0 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ 0: 𝑥𝑥 =
2𝑥𝑥 + 𝑎𝑎𝑎𝑎 = 𝑎𝑎
3𝑥𝑥 − 𝑎𝑎𝑎𝑎 = 5
𝑥𝑥
𝑦𝑦
1
+
= 2
2
�𝑎𝑎 − 𝑏𝑏 𝑎𝑎 + 𝑏𝑏 𝑎𝑎 − 𝑏𝑏
𝑥𝑥 𝑦𝑦
1
+ =
𝑎𝑎 𝑏𝑏 𝑎𝑎𝑎𝑎
𝑎𝑎 ≠ ±𝑏𝑏 ∧ 𝑎𝑎 ≠ 0 ∧ 𝑏𝑏 ≠ 0
𝑏𝑏
𝑎𝑎
𝑥𝑥 = 2
, 𝑦𝑦 = 2
;
𝑎𝑎 + 𝑏𝑏 2
𝑎𝑎 + 𝑏𝑏 2
𝑥𝑥 + 𝑦𝑦 + 3𝑎𝑎
𝑦𝑦
2𝑎𝑎
+
= 2
𝑎𝑎 + 1 𝑎𝑎 − 1
� 4𝑎𝑎 − 4
𝑥𝑥
𝑦𝑦
𝑎𝑎 + 2
+
=
𝑎𝑎 𝑎𝑎 + 1 𝑎𝑎 + 1
𝑎𝑎 ≠ 0 ∧ 𝑎𝑎 ≠ ±1
3
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ : 𝑥𝑥 = 2𝑎𝑎, 𝑦𝑦 = −𝑎𝑎;
4
3
𝑠𝑠𝑠𝑠 𝑎𝑎 = 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
4
𝑎𝑎(𝑥𝑥 − 𝑦𝑦 − 2) = 𝑥𝑥 + 3𝑦𝑦 + 7
𝑦𝑦 + 2
1
� 𝑥𝑥
+
= 2
𝑎𝑎 + 3 1 − 𝑎𝑎 𝑎𝑎 + 2𝑎𝑎 − 3
𝑎𝑎 ≠ −3 ∧ 𝑎𝑎 ≠ 1 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝑥𝑥 + 2𝑎𝑎 𝑦𝑦 + 𝑏𝑏
+
=7
𝑏𝑏
� 𝑎𝑎
2𝑥𝑥 − 𝑎𝑎 3𝑦𝑦 − 𝑏𝑏
−
= −2
𝑎𝑎
𝑏𝑏
𝑎𝑎 ≠ 0 ∧ 𝑏𝑏 ≠ 0 𝑥𝑥 = 2𝑎𝑎, 𝑦𝑦 = 2𝑏𝑏
𝑎𝑎𝑎𝑎
𝑦𝑦
− =1
𝑏𝑏(𝑎𝑎 + 𝑏𝑏) 𝑏𝑏
�
2𝑥𝑥
𝑦𝑦 − 𝑎𝑎 + 𝑏𝑏 2(𝑎𝑎 + 𝑏𝑏)
−
=
𝑎𝑎 − 𝑏𝑏
𝑏𝑏
𝑎𝑎 − 𝑏𝑏
𝑎𝑎 ≠ ±𝑏𝑏 ∧ 𝑎𝑎 ≠ 0 ∧ 𝑏𝑏 ≠ 0
1
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ − : 𝑥𝑥 = 1, 𝑦𝑦 = 𝑎𝑎;
5
1
𝑠𝑠𝑠𝑠 𝑎𝑎 = − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
5
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Sistemi di equazioni di I grado letterali con discussione
1
: 𝑥𝑥 = −𝑎𝑎, 𝑦𝑦 = 𝑎𝑎;
2
1
𝑠𝑠𝑠𝑠 𝑎𝑎 = 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
2
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠
𝑎𝑎𝑎𝑎 + (𝑎𝑎 + 1)𝑦𝑦 = 𝑎𝑎
�
𝑥𝑥 + 3𝑦𝑦 = 2𝑎𝑎
�
𝑎𝑎𝑎𝑎 + 2𝑦𝑦 = 4
(𝑎𝑎 − 1)𝑥𝑥 + 𝑦𝑦 = 𝑎𝑎
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ 2: 𝑥𝑥 = 2, 𝑦𝑦 = 2 − 𝑎𝑎;
𝑠𝑠𝑠𝑠 𝑎𝑎 = 2 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝑎𝑎 + 𝑥𝑥
3
1
− 𝑦𝑦 −
= −2 +
2𝑎𝑎
2𝑎𝑎
� 𝑎𝑎
𝑥𝑥 − 𝑎𝑎 𝑦𝑦 + 𝑎𝑎 13 + 𝑎𝑎
+
=
3
2
6
𝑎𝑎 ≠ 0
3
𝑥𝑥 = 2, 𝑦𝑦 = 3
2
3
𝑠𝑠𝑠𝑠 𝑎𝑎 = − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
2
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ −
𝑥𝑥 + 𝑦𝑦 𝑥𝑥 − 𝑦𝑦 2(𝑎𝑎2 + 𝑏𝑏 2 )
⎧
+
=
𝑏𝑏
𝑎𝑎
𝑎𝑎𝑎𝑎
𝑥𝑥
+
𝑦𝑦
𝑎𝑎
⎨
=
⎩ 𝑥𝑥 − 𝑦𝑦 𝑏𝑏
𝑎𝑎 ≠ 0 ∧ 𝑏𝑏 ≠ 0:
𝑥𝑥 = 𝑎𝑎 + 𝑏𝑏, 𝑦𝑦 = 𝑎𝑎 − 𝑏𝑏
𝑥𝑥 + 𝑦𝑦 𝑥𝑥 − 𝑦𝑦
+
=4
� 𝑎𝑎
𝑏𝑏
𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏 = (𝑎𝑎 + 𝑏𝑏)2 − 2𝑏𝑏 2
𝑎𝑎 ≠ 0 ∧ 𝑏𝑏 ≠ 0
𝑥𝑥 = 𝑎𝑎 + 𝑏𝑏, 𝑦𝑦 = 𝑎𝑎 − 𝑏𝑏
2𝑏𝑏 + 𝑎𝑎
𝑎𝑎
�
2𝑎𝑎 − 𝑏𝑏
2𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏 =
𝑎𝑎
2𝑏𝑏𝑏𝑏 − 𝑎𝑎𝑎𝑎 =
𝑎𝑎 ≠ 0
1
1
𝑥𝑥 = , 𝑦𝑦 = −
𝑎𝑎
𝑎𝑎
𝑏𝑏(𝑥𝑥 + 1) 𝑎𝑎(𝑦𝑦 − 1)
+
= 𝑎𝑎 + 𝑏𝑏
�
𝑎𝑎
𝑏𝑏
𝑎𝑎(𝑥𝑥 − 𝑎𝑎) − 𝑏𝑏(𝑦𝑦 − 𝑏𝑏) = −(𝑎𝑎 + 𝑏𝑏)
𝑎𝑎 ≠ 0 ∧ 𝑏𝑏 ≠ 0
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ −𝑏𝑏: 𝑥𝑥 = 𝑎𝑎 − 1, 𝑦𝑦 = 𝑏𝑏 + 1;
𝑠𝑠𝑠𝑠 𝑎𝑎 = −𝑏𝑏 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝑥𝑥 − 𝑦𝑦 𝑥𝑥 − 𝑦𝑦
+
= 2𝑎𝑎
𝑎𝑎 + 𝑏𝑏 𝑎𝑎 − 𝑏𝑏
�3(𝑥𝑥 − 𝑦𝑦) 5(𝑥𝑥 − 𝑦𝑦)
+
= 2(4𝑎𝑎 + 𝑏𝑏)
𝑎𝑎 + 𝑏𝑏
𝑎𝑎 − 𝑏𝑏
𝑎𝑎 ≠ ±𝑏𝑏
𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝑥𝑥 − 𝑦𝑦 + 𝑎𝑎 𝑦𝑦 − 𝑥𝑥 + 𝑏𝑏 9𝑎𝑎2 − 2𝑎𝑎𝑎𝑎 − 𝑏𝑏 2
� 𝑎𝑎 + 𝑏𝑏 − 𝑎𝑎 − 𝑏𝑏 =
𝑎𝑎2 − 𝑏𝑏 2
𝑥𝑥 − 2𝑎𝑎 = −𝑦𝑦
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𝑎𝑎 ≠ ±𝑏𝑏
𝑠𝑠𝑠𝑠 𝑎𝑎 ≠ 0 𝑥𝑥 = 3𝑎𝑎, 𝑦𝑦 = −𝑎𝑎;
𝑠𝑠𝑠𝑠 𝑎𝑎 = 0 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
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