32 Algebraik kasrlarni qo’shish va ayirish(5 soat)

Author:

October 21st, 2022

7-sinf, algebra.
(5 soatlik bayonnoma)
«_______»-aprel, 201___-yil.
Mavzu: Algebraik kasrlarni qo‘shish va ayirish.
1. Darsning maqsadi: umumiy ko‘paytuvchini qavsdan tashqariga chiqarish, algebraik kasrlarni
qisqartirish, kasrni umumiy maxrajga keltirish, algebraik kasrlarni qo‘shish va ayirish kabi
mavzular yuzasidan tushunchalar hosil qilish, o‘quvchilarning BKM larini shakllantirish,
rivojlantirish va mustahkamlash.
2. Darsning usuli: savol-javob, misol va masalalar yechish.
3. Darsning jihozi: DTS ko‘rgazmalari, tarqatma materiallar, kichik testlar.
DARSNING BORISHI:
1. Tashkiliy qism: o‘quvchilar bilan salomlashish, tozalikni tekshirish, davomatni aniqlash,
o‘quvchilarning dars mashg‘ulotlariga ruhiy jihatda tayyorliklarini aniqlash.
2. O‘tilgan mavzuni so‘rab baholash:
Savol: Kasrlarni qisqartirish deb nimaga aytiladi?
Javob: Kasrning surat va maxrajini ularning umumiy ko‘paytuvchisiga bo‘lish kasrni qisqartirish
deyiladi.
Savol: Algebraik kasr deb nimaga aytiladi?
Javob: Surat va maxraji algebraik ifodalar bo‘lgan kasr algebraik kasr deyiladi.
Savol: Kasrning asosiy xossasi nimadan iborat?
Javob: Kasrning surat va maxraji ayni bir xil songa ko‘paytirilsa yoki bo‘linsa, unga teng kasr
hosil bo‘ladi.
Savol: Kasrni umumiy maxrajga keltirish nima?
Javob: Kasrning surat va maxrajini ularning EKUK iga ko‘paytirish – umumiy maxrajga keltirish
deyiladi.
3. O‘tilgan mavzuni mustahkamlasi: o‘quvchilar tushunmagan savollar va ularga tushunarsiz
bo‘lgan jumlalar aniq va hayotiy misollar yordamida tushunturilib beriladi.
4. Yangi mavzuning bayoni:
Algebraik kasrlarni qo‘shish va ayirish oddiy kasrlarni qo‘shish va ayirish kabi bajariladi.
Ta’rif: bir xil maxrajli kasrlarni qo‘shish va ayirish uchun maxrajlardan biri yozib olinadi,
so‘ngra kasrning suratlari qo‘shiladi (ayriladi).
𝑎 𝑐 𝑎+𝑐 𝑚 𝑝 𝑚−𝑝
+ =
;
− =
.
𝑏 𝑏
𝑏
𝑛 𝑛
𝑛
Ta’rif: har xil maxrajli kasrlarni qo‘shish va ayirish uchun – ular bir xil maxrajga keltiriladi,
so‘ngra, bir xil maxrajli kasrlarni qo‘shish va ayirish kabi bajariladi.
𝑑/
𝑏/
𝑞/
𝑛/
𝑎
𝑐 𝑎𝑑 + 𝑏𝑐
𝑚
𝑝 𝑚𝑞 − 𝑛𝑝
𝑎 𝑐
𝑚 𝑝
+ =
+
=
;
− =
−
=
.
𝑏 𝑑
𝑏
𝑑
𝑏𝑑
𝑛 𝑞
𝑛
𝑞
𝑛𝑞
Umuman, turli maxrajli kasrlarni qo‘shish va ayirish uchun quyidagi tartibga amal qilish
kerak:
1) kasrlarning umumiy maxraji topiladi;
2) kasrlar umumiy maxrajga keltiriladi;
3) hosil bo‘lgan kasrlar qo‘shiladi;
4) mumkin bo‘lsa, natija soddalashtiriladi.
Misol va masalalar yechish:
485-misol. Kasrlarning yig‘indisi (ayirmasi) ni toping:
Yechilishi:
𝑝
3𝑝
𝑝+3𝑝
4𝑝
8𝑎
3𝑎
8𝑎−3𝑎
5𝑎
𝑎
𝑐
𝑎+𝑐
1) 𝑞2 + 𝑞2 = 𝑞2 = 𝑞2 ; 2) 𝑏3 − 𝑏3 = 𝑏3 = 𝑏3 ; 3) 𝑎+𝑏 + 𝑎+𝑏 = 𝑎+𝑏; 4)
𝑥
𝑦
𝑥−𝑦
− 𝑛+𝑎 = 𝑛+𝑎.
𝑛+𝑎
486-misol. Kasrlarning yig‘indisi (ayirmasi) ni toping:
Yechilishi:
𝑐+𝑑
2𝑐−𝑑
𝑐+𝑑+2𝑐−𝑑
3𝑐
𝑎+2𝑏
5𝑎−2𝑏
𝑎+2𝑏+5𝑎−2𝑏
6𝑎
2𝑎
1) 2𝑎 + 2𝑎 =
=
;
2)
+
=
=
=
;
2
2
2
2
2𝑎
2𝑎
3𝑐
3𝑐
3𝑐
3𝑐
𝑐2
3)
5)
6)
𝑎+𝑏
2𝑐
𝑎−𝑏
−
(1+𝑏)2
(2+𝑎)2
(2−𝑎)2
−
𝑎2 𝑏
𝑏
= 𝑐;
4)
(2+𝑎)2 −(2−𝑎)2
3𝑎−𝑏
−
𝑎3
4+4𝑎+𝑎2 −4+4𝑎−𝑎2
10𝑎−𝑏−3𝑎+𝑏
=
𝑎3
𝑎3
1+𝑏 2 +1+𝑏2
=
5𝑑
=
𝑎2 𝑏
10𝑎−𝑏
1+2𝑏+𝑏2 +1−2𝑏+𝑏 2
=
5𝑑
=
𝑎2 𝑏
2𝑐
(1+𝑏)2 +(1−𝑏)2
=
5𝑑
2𝑏
=
2𝑐
(1−𝑏)2
+
5𝑑
𝑎+𝑏−𝑎+𝑏
=
2𝑐
8𝑎
2+2𝑏 2
=
5𝑑
=
7𝑎
𝑎3
=
5𝑑
;
2(1+𝑏2 )
5𝑑
;
8
= 𝑎2 𝑏 = 𝑎𝑏.
𝑎2 𝑏
487-misol. Kasrlarning yig‘indisi (ayirmasi) ni toping:
Yechilishi:
7/
1)
3
+
5
5/
4)
5/
2
14+15
=
7
2
−
𝑏
= 35;
35
1/
1
5−2
=
5𝑏
4/
29
2)
7
1/
3
= 5𝑏;
5𝑏
4
5)
1/
−
5
=
28
5𝑎/
𝑐
+
15𝑎
𝑑
3
16−5
= 28;
28
=
1/
11
𝑐+5𝑎𝑑
15𝑎
3)
3𝑑/
1
+
3𝑎
6)
;
3/
2
2+3
=
𝑎
3𝑎
1/
𝑎
𝑏
−
4
3𝑎𝑑−𝑏
=
12𝑑
5
= 3𝑎;
12𝑑
.
489-misol. Kasrlarning yig‘indisi (ayirmasi) ni toping:
Yechilishi:
1)
3)
𝑏2/
𝑏/
2
5−
𝑑2 /
𝑏
1/
3
+
𝑏2
𝑑/
𝑐
𝑑−
𝑑
=
1/ 2
𝑐
+
=
𝑑2
𝑐/
5𝑏 2 −2𝑏+3
2)
;
𝑏2
𝑑3 −𝑑𝑐+𝑐 2
𝑑2
2
𝑐
𝑛/
4)
;
+
𝑚
𝑐 2/
−
𝑛
1/
3
4−
𝑛2 /
2𝑐+4𝑐 2 −3
=
𝑐2
1/
𝑚2
𝑘+
𝑚𝑛−𝑛2 𝑘+𝑚2
=
𝑛2
;
𝑐2
𝑛2
.
490-misol. Kasrlarning yig‘indisi (ayirmasi) ni toping:
Yechilishi:
𝑐/
1)
𝑎/
1
𝑑/
4)
1
+
𝑎𝑏
𝑎/
𝑏
𝑏
+
𝑎𝑐
=
𝑐𝑑
𝑘/
𝑐+𝑎
=
𝑏𝑐
2)
;
𝑎𝑏𝑐
𝑏𝑑+𝑎𝑑
𝑎𝑐𝑑
=
−
𝑚𝑛
𝑛/
𝑑(𝑏+𝑎)
𝑎𝑐𝑑
1
;
5)
3
+
𝑚2
𝑛/
1
𝑑/
𝑘−𝑛
= 𝑚𝑛𝑘;
𝑚𝑘
𝑚/
4
=
𝑚𝑛
3𝑛+4𝑚
𝑚2
−
𝑏𝑐
𝑛2 /
6)
;
𝑐/
𝑎
3)
𝑎
𝑏𝑑
=
𝑚/
2
−
𝑚𝑛
𝑎𝑑−𝑎𝑐
𝑏𝑐𝑑
3
𝑛3
=
𝑎(𝑑−𝑐)
=
𝑏𝑐𝑑
2𝑛2 −3𝑚
𝑚𝑛3
;
.
491-misol. Kasrlarning yig‘indisi (ayirmasi) ni toping:
Yechilishi:
3𝑏2 /
1)
2𝑎2 /
3𝑐
4𝑎3 𝑏
2
3𝑦
3 −
𝑎 2 𝑐2 /
5)
9𝑏 2 𝑐+10𝑎2 𝑑
12𝑎3 𝑏 3
𝑥𝑦/
1
6𝑥 2 𝑦
𝑎2 𝑏2 /
𝑎
𝑏
6𝑎𝑏
3 =
2𝑦2 /
4𝑥2 /
3)
5𝑑
+
2 +
𝑏
𝑐
+
5
12𝑥𝑦 2
𝑏2 𝑐2 /
2 +
𝑐
𝑎2
=
=
2𝑎3 /
;
2)
8𝑥 2 −2𝑦 2 +5𝑥𝑦
12𝑥 2 𝑦 3
𝑎3 𝑐 2 +𝑎2 𝑏 3 +𝑏 2 𝑐 3
𝑎2 𝑏 2 𝑐 2
9𝑏
4𝑦/
;
4)
3𝑏3 /
2𝑎
6)
7𝑥 2 𝑦
𝑏
𝑐
=
6𝑎3 𝑏
4𝑎4 −21𝑏 3 𝑐
18𝑎3 𝑏 4
7𝑥/
5
3
−
𝑐𝑑2 /
;
7𝑐
4 −
4𝑥𝑦
2/
11
2 +
𝑑/
+
14𝑥 2 𝑦 2
𝑐/
𝑏
𝑐2𝑑
;
𝑏
+
𝑐𝑑 2
=
=
20𝑦−21𝑥+22
28𝑥 2 𝑦 2
𝑏𝑐𝑑 2 +𝑏𝑑+𝑏𝑐
𝑐 2 𝑑2
=
;
𝑏(𝑐𝑑 2 +𝑑+𝑐)
𝑐 2𝑑2
;
493-misol. Algebraik kasrlarni qo‘shing va ayiring:
Yechilishi:
2/
1)
5
2𝑥−2
2/
2)
1/
+
𝑎
3𝑎+3𝑏
3
4𝑥−4
1/
2𝑎
10+3
2/
13
= 4𝑥−4 = 4(𝑥−1);
2𝑎−2𝑎
0
2)
− 6𝑎+6𝑏 = 6𝑎+6𝑏 = 6𝑎+6𝑏 − ma’noga ega emas; 4)
7
5𝑏+5
1/
−
2/
3𝑥
4𝑥+4𝑦
3
10𝑏+10
1/
𝑥
14−3
11
= 10𝑏+10 = 10(𝑏+1);
6𝑥−𝑥
5𝑥
− 8𝑥+8𝑦 = 8𝑥+8𝑦 = 8(𝑥+𝑦);
494-misol. Algebraik kasrlarni qo‘shing va ayiring:
Yechilishi:
𝑏/
5𝑎
𝑎𝑏+𝑏
𝑎/
3
5𝑎
3𝑏+5𝑎 2
;
𝑎𝑏(𝑎+1)
1)
3
𝑎 2 +𝑎
3)
𝑦+𝑎
𝑏2 +𝑏𝑎
+ 𝑎𝑏+𝑎2 = 𝑏(𝑏+𝑎) + 𝑎(𝑏+𝑎) =
4)
𝑦−𝑏
𝑎 2 −𝑎𝑏
−
+
=
+
𝑎(𝑎+1)
𝑎/
𝑦−𝑏
𝑏(𝑎+1)
𝑏/
𝑦+𝑎
𝑦−𝑏
𝑏/
𝑦−𝑎
𝑎𝑏−𝑏2
−
𝑎(𝑎−𝑏)
𝑦−𝑎
5𝑏
𝑎𝑥+𝑎𝑦
−
2𝑎
𝑏𝑥+𝑏𝑦
𝑏/
=
𝑎/
5𝑏
𝑎(𝑥+𝑦)
−
2𝑎
𝑏(𝑥+𝑦)
=
5𝑏2 −2𝑎2
;
𝑎𝑏(𝑥+𝑦)
𝑏(𝑦−𝑏)−𝑎(𝑦−𝑎)
.
𝑎𝑏(𝑎−𝑏)
=
𝑏(𝑎−𝑏)
2)
𝑎(𝑦+𝑎)+𝑏(𝑦−𝑏)
;
𝑎𝑏(𝑏+𝑎)
𝑎/
𝑦−𝑏
=
=
495-misol. Algebraik kasrlarni qo‘shing va ayiring:
Yechilishi:
𝑥/
1)
𝑥+𝑦/
3
−
𝑥+𝑦
𝑎−1/
2)
6
𝑎
𝑎/
−
𝑥+3/
3)
5
3𝑥−5(𝑥+𝑦)
=
𝑥
10
𝑥(𝑥+𝑦)
6𝑎−6−10𝑎
=
𝑎−1
𝑥−3/
1
+
𝑥(𝑥−3)
8(𝑎+𝑏)/
4)
=
𝑎(𝑎−1)
1
−4𝑎−6
5(𝑎−𝑏)
8(𝑎+𝑏)
−2𝑥−5𝑦
−2(2𝑎+3)
2𝑥
;
𝑎(1−𝑎)
;
2
32𝑎+32𝑏−35𝑎+35𝑏
40(𝑎2 −𝑏 2 )
=
𝑥(𝑥+𝑦)
2(2𝑎+3)
=
−𝑎(1−𝑎)
−(2𝑥+5𝑦)
=
𝑥(𝑥+𝑦)
= 𝑥(𝑥 2−9) = 𝑥 2−9;
𝑥(𝑥 2 −9)
7
−
=
−𝑎(1−𝑎)
5(𝑎−𝑏)/
4
=
𝑥(𝑥+𝑦)
𝑥+3+𝑥−3
=
𝑥(𝑥+3)
3𝑥−5𝑥−5𝑦
=
−3𝑎+67𝑏
=
67𝑏−3𝑎
=
40(𝑎2 −𝑏 2 ) 40(𝑎2 −𝑏2 )
.
497-misol. Algebraik kasrlarni qo‘shing va ayiring:
Yechilishi:
𝑥+4/
1)
3)
−
𝑥−4
5𝑥−2
16−𝑥2
2𝑐−3/ 2
𝑐 −8
21𝑦 2 +1
1−9𝑦 2
2𝑥 2 +8𝑥+5𝑥−2
=
16𝑐−2𝑐 3
1+3𝑦/
−
𝑦
3𝑦−1
4𝑐 2 −9
21𝑦 2 +1+𝑦+3𝑦 2
=
𝑥 2 −16
2)
;
2𝑐 3 −16𝑐−3𝑐 2 +24+16𝑐−2𝑐 3
=
9−4𝑐 2
1/
2𝑥 2 +13𝑥−2
=
𝑥 2 −16
1/
−
2𝑐+3
1/
4)
1/
2𝑥
1−9𝑦 2
24𝑦 2 +𝑦+1
=
1−9𝑦 2
=
𝑛+7/
12𝑛−5
6
+
𝑛2 −49
−3𝑐 2 +24
4𝑐 2 −9
=
7−𝑛
−3(𝑐 2 −8)
4𝑐 2 −9
=
12𝑛−5−6𝑛−42
=−
𝑛2 −49
3(𝑐 2 −8)
4𝑐 2 −9
=
6𝑛−47
𝑛2 −49
;
;
.
498-misol. Algebraik kasrlarni qo‘shing va ayiring:
Yechilishi:
𝑎+2/
1)
1/
3
2𝑎
+ (𝑎+2)2 =
𝑎+2
3𝑎+6+2𝑎
(𝑎+2)2
1/
5𝑎+6
= (𝑎+2)2;
2)
3𝑎+1/
𝑎
+
(3𝑎+1)2
4
=
3𝑎+1
𝑎+12𝑎+4
(3𝑎+1)2
13𝑎+4
= (3𝑎+1)2.
499-misol. Algebraik kasrlarni qo‘shing va ayiring:
Yechilishi:
1)
2𝑦+8
1/
2𝑦+8
7
− 𝑦−2 =
𝑦 2 −4𝑦+4
4−5𝑥
(𝑦−2)2
𝑦−2/
−
𝑦−2
1/
2)
− 3𝑥+1 = (3𝑥+1)2 −
1+6𝑥+9𝑥 2
3)
7
(𝑎−𝑏)2
4)
4
(𝑚−𝑛)2
5)
1/
5
− 𝑏−𝑎 =
+
(𝑎−𝑏)2
1/
7
𝑚−𝑛/
4
7
=
𝑚−𝑛
𝑎+5/
10
+ 𝑎2 −25 =
25−10𝑎+𝑎2
2(𝑎−5)2
5
=
𝑎−𝑏
− 𝑛−𝑚 = (𝑚−𝑛)2 +
2𝑎
2
3𝑥+1
𝑎−𝑏/
7
2𝑦+8−𝑦+2
(𝑦−2)2
=
3𝑥+1/
4−5𝑥
2
7
4−5𝑥−6𝑥−2
(3𝑥+1)2
=
2−11𝑥
= (3𝑥+1)2;
7+5(𝑎−𝑏)
;
(𝑎−𝑏)2
4+7(𝑚−𝑛)
;
(𝑚−𝑛)2
𝑎−5/
2𝑎
𝑦+10
= (𝑦−2)2;
10
+ (𝑎−5)(𝑎+5) =
(𝑎−5)2
2𝑎2 +10𝑎+10𝑎−50
(𝑎+5)(𝑎−5)2
=
2𝑎2 +20𝑎−50
(𝑎+5)(𝑎−5)2
=
2(𝑎2 +10𝑎−25)
(𝑎+5)(𝑎−5)2
2
= (𝑎+5)(𝑎−5)2 = 𝑎+5;
6)
1
𝑥 2 −6𝑥+9
+
1
(𝑥+3)2
(𝑥+3)2 /
=
1
(𝑥−3)2
(𝑥−3)2 /
+
1
(𝑥+3)2
=
(𝑥+3)2 +(𝑥−3)2
(𝑥 2 −9)2
=
𝑥 2 +6𝑥+9+𝑥 2 −6𝑥+9
(𝑥 2 −9)2
=
=
2(𝑥 2 +9)
2𝑥 2 +18
= (𝑥 2 −9)2 =
(𝑥 2 −9)2
.
501-misol. Algebraik kasrlarni qo‘shing va ayiring:
Yechilishi:
𝑎−𝑏/
1)
3)
4)
1/
8
+
𝑎+𝑏
1/
2)
𝑎+𝑏/
7
𝑥+𝑦/
6𝑥
𝑥−𝑦/
3
−
𝑥 2 −𝑦 2
3
16𝑏
7𝑎−7𝑏+8𝑎+8𝑏−16𝑏
− 𝑎2 −𝑏2 =
𝑎−𝑏
−
𝑥−𝑦
2
4
𝑎−3/
6
+ 3−𝑎 − 𝑎2−9 =
𝑎+3
3
8
6𝑥−3𝑥−3𝑦−4𝑥+4𝑦
=
𝑥+𝑦
𝑎+3/
1/
2𝑎−3/
3
7
6
𝑎2 −9
2𝑎+3/
8
15
−(𝑥−𝑦)
3𝑎−9−2𝑎−6−6
− 𝑎2−9 =
𝑎−3
− 2𝑎+3 − 3−2𝑎 = 4𝑎2 −9 −
4𝑎2 −9
15(𝑎−𝑏)
= (𝑎+𝑏)(𝑎−𝑏) = 𝑎+𝑏;
−𝑥+𝑦
1/
2
−
𝑎+3
𝑎2 −𝑏 2
−1
1
= 𝑥 2 −𝑦 2 = (𝑥−𝑦)(𝑥+𝑦) = 𝑥+𝑦 = − 𝑥+𝑦;
𝑥 2 −𝑦 2
3
15𝑎−15𝑏
=
𝑎2 −𝑏 2
7
+
2𝑎+3
𝑎−21
𝑎2 −9
;
3−16𝑎+24+14𝑎+21
=
2𝑎−3
=
4𝑎2 −9
=
−2𝑎+48
4𝑎2 −9
=
2(24−𝑎)
4𝑎2 −9
.
502-misol. Algebraik kasrlarni qo‘shing va ayiring:
Yechilishi:
𝑎−𝑏/
1)
2)
=
3)
𝑎/
𝑎+𝑏
−
𝑎
5𝑏−1
𝑏
− 𝑎2−𝑎𝑏 =
𝑎−𝑏
𝑏+2
𝑎2 −𝑏 2 −𝑎2 −𝑏
𝑎2 −𝑎𝑏
2/
3(𝑏−1)/
5𝑏−1
𝑏+1
+ 2𝑏+2 − 𝑏−1 = 3(𝑏2−1) +
3𝑏 2 −3
𝑏−14−3𝑏2
6(𝑏 2 −1)
6𝑎
3𝑎+1
36𝑎
6/
2(9𝑎2 +6𝑎+1)
+
6(9𝑎2 −1)
4
3(9𝑎2 −6𝑎+1)
=
6(9𝑎2 −1)
𝑏−1
=
𝑏(𝑏+1)
= − 𝑎(𝑎−𝑏);
10𝑏−2+3𝑏2 +6𝑏−3𝑏−6−6𝑏2 −6𝑏−6𝑏−6
6(𝑏2 −1)
(𝑥−𝑦)/
𝑥 2 −𝑦 2 /
𝑥−
=
− 𝑥 2 −𝑦 2 =
𝑥+𝑦
4𝑎
𝑎3 +𝑏
6) 𝑎 − 2 + 2+𝑎 − 𝑎2 +2𝑎 =
4𝑎2 −4𝑎−𝑏
2(3𝑎+1)
6(9𝑎2 −1)
𝑚/
𝑚−𝑛
4
𝑎−
𝑎/
𝑎(𝑎+2)/
2+
3(3𝑎−1)2
𝑚
−𝑥 2 𝑦
=
9𝑎2 +6𝑎+1
6(9𝑎2 −1)
+
;
4𝑚(2𝑛+𝑚)
2𝑛−𝑚
𝑚(𝑚−𝑛)
− 4𝑛2 −𝑚2 =
28𝑛2 −4𝑚2 +9𝑚𝑛
;
𝑚(4𝑛2 −𝑚2 )
𝑥2𝑦
= 𝑥 2 −𝑦 2 = − 𝑥 2 −𝑦 2;
𝑥 2 −𝑦 2
𝑎(𝑎+2)/
=
7(4𝑛2 −𝑚2 )
28𝑛2 −7𝑚2 +8𝑚𝑛+4𝑚2 −𝑚2 +𝑚𝑛
𝑚(4𝑛2 −𝑚2 )
𝑥 3 −𝑥𝑦 2 −𝑥 2 𝑦+𝑥𝑦 2 −𝑥 3
2(3𝑎+1)2
36𝑎
= 6(9𝑎2−1) − 6(9𝑎2−1) + 6(9𝑎2−1) =
− 4𝑛2 −𝑚2 =
2𝑛−𝑚
𝑚2 −𝑚𝑛
1/ 3
𝑥
𝑥𝑦
3𝑎−1
36𝑎−18𝑎2 −12𝑎−2+27𝑎2 −18𝑎+3
+
𝑚
8𝑚𝑛+4𝑚2
3(3𝑎−1)/
3𝑎+1
𝑚(2𝑛+𝑚).
7
− 𝑚−2𝑛 − 4𝑛2 −𝑚2 =
𝑚
𝑎2 +2𝑎
𝑎(𝑎−𝑏)
𝑏+1
+
3(3𝑎−1)
4𝑛2 −𝑚2 /
𝑚−𝑛
28𝑛2 −7𝑚2
=
−𝑏(𝑏+1)
=
6(𝑏+1)/
𝑏+2
2(3𝑎+1)/
6𝑎
3𝑎−1
= 𝑚(4𝑛2−𝑚2 ) + 𝑚(4𝑛2−𝑚2 ) − 𝑚(4𝑛2−𝑚2) =
5)
𝑎2 −𝑎𝑏
−
2(𝑏+1)
+ 3−9𝑎 + 6𝑎+2 = 9𝑎2 −1 −
9𝑎2 −1
7
−𝑏 2 −𝑏
=
;
= 6(9𝑎2 −1) −
4)
1/
𝑎
4𝑎
𝑎+2
1/ 3
𝑎 +𝑏
− 𝑎(𝑎+2) =
𝑎3 +2𝑎2 −2𝑎2 −4𝑎+4𝑎2 −𝑎3 −𝑏
𝑎2 +2𝑎
=
.
503-misol. Algebraik kasrlarni qo‘shing va ayiring:
Yechilishi:
1)
2)
𝑎+1
𝑎3 −1
− 𝑎2+𝑎+1 =
1/ 2
𝑎 +4
𝑎3 +8
4)
=
𝑎+2
𝑎+𝑏
𝑚3 −27
−
𝑎3 +8
(𝑎2 −𝑎𝑏+𝑏2 )
1
=
𝑎+𝑏
(𝑚2 +3𝑚+9)
𝑎+1−𝑎+1
𝑎3 −1
𝑎2 +4−𝑎2 +2𝑎−4
−
𝑎2 −𝑎𝑏+𝑏 2
𝑚2 −3𝑚+9
1
− 𝑎2+𝑎+1 =
𝑎3 −1
1
−
(𝑎−1)/
𝑎+1
(𝑎2 −2𝑎+4)/
(𝑎+𝑏)/
3)
1/
1
1
=
𝑚−3
2
= 𝑎3 −1;
2𝑎
= 𝑎3 +8;
(𝑎+𝑏)2 −𝑎2 +𝑎𝑏−𝑏 2
𝑎3 +𝑏 3
𝑚2 −3𝑚+9−𝑚2 −3𝑚−9
𝑚3 −27
=
=
𝑎2 +𝑎𝑏+𝑏 2 −𝑎2 +𝑎𝑏−𝑏 2
𝑎3 +𝑏 3
−3𝑚−3𝑚
𝑚3 −27
−6𝑚
2𝑎𝑏
= 𝑎3 +𝑏3;
6𝑚
= 𝑚3 −27 = − 𝑚3 −27.
5. O‘tilgan mavzuni mustahkamlash: o‘quvchilar tushunmagan savollarni aniq misollar yordamida
tushuntiraman.
6. O‘tilgan mavzuni so‘rab baholash:
Savol: Algebraik kasr deb nimaga aytiladi?
Javob: Surat va maxraji algebraik ifodalar bo‘lgan kasr algebraik kasr deyiladi.
Savol: Kasrning asosiy xossasi nimadan iborat?
Javob: Kasrning surat va maxraji ayni bir xil songa ko‘paytirilsa yoki bo‘linsa, unga teng
kasr hosil bo‘ladi.
Savol: Kasrni umumiy maxrajga keltirish nima?
Javob: Kasrning surat va maxrajini ularning EKUK iga ko‘paytirish – umumiy maxrajga
keltirish deyiladi.
Savol: Bir xil maxrajli kasrlar qanday qo‘shiladi (ayiriladi)?
Javob: Bir xil maxrajli kasrlarni qo‘shish (ayirish) uchun maxrajlardan biri yozib olinadi,
suratlari esa qo‘shiladi (ayiriladi).
Savol: Har xil maxrajli kasrlarni qo‘shish (ayirish) qanday bajariladi?
Javob: Har xil maxrajli kasrlarni qo‘shish (ayirish) uchun, ular bir xil maxrajga keltiriladi,
so‘ngra bir xil maxrajli kasrlarni qo‘shish (ayirish) kabi bajariladi.
7. Uyga vazifa: O‘tilgan mavzuni o‘qib o‘rganish va misollar yechish.
488-misol. Kasrlarning yig‘indisi (ayirmasi) ni toping:
Yechilishi:
𝑛/
1)
2/
𝑚
1
−
2
𝑛
5/
𝑚𝑛−2
=
; 2)
2𝑛
𝑎/
3
+
𝑎
𝑏
=
5
15+𝑎𝑏
5𝑎
; 3)
𝑎/
1/
1
5−
=
𝑎
1/
2
5𝑎−1
; 4)
𝑎
𝑏/
+
𝑏
7=
2+7𝑏
𝑏
.
492-misol. Algebraik kasrlarni qo‘shing va ayiring:
Yechilishi:
1/
1)
𝑥
+
3(𝑎−𝑏)
4/
3)
3/
2𝑥
3/
2𝑎2
3(𝑎+1)
+
2𝑥+3𝑥
1/
5𝑥
= 3(𝑎−𝑏) = 3(𝑎−𝑏);
𝑎−𝑏
5𝑎2
8𝑎2 +15𝑎2
=
4(𝑎+1)
12(𝑎+1)
=
2)
2/
7𝑥
5𝑥
−
2(𝑥−1)
𝑥−1
2/
23𝑎2
4)
;
12(𝑎+1)
5/
4𝑦
5(𝑦−3)
−
7𝑥−10𝑥
=
5𝑥
2(𝑦−3)
2(𝑥−1)
8𝑦−25𝑥
=
−3𝑥
= 2(𝑥−1);
.
10(𝑦−3)
496-misol. Algebraik kasrlarni qo‘shing va ayiring:
Yechilishi:
1/
1)
+
1−𝑏2
1/
3)
1−𝑏/
𝑎
5+𝑝2
𝑝2 −36
1
=
1+𝑏
𝑝−6/
−
𝑎+1−𝑏
1−𝑏2
𝑝
=
6+𝑝
=
1/
𝑎−𝑏+1
1−𝑏 2
5+𝑝2 −𝑝2 +6𝑝
2)
;
+
𝑥 2 −9
𝑥+4/
5+6𝑝
= 𝑝2−36; 4)
𝑝2 −36
𝑥−3/
2
1
𝑥+3
=
2+𝑥−3
1/
2𝑥
5𝑥−2
−
𝑥−4
=
𝑥 2 −16
𝑥−1
= 𝑥 2 −9;
𝑥 2 −9
2𝑥 2 +8𝑥−5𝑥+2
𝑥 2 −16
=
2𝑥 2 +3𝑥+2
𝑥 2 −16
.
500-misol. Algebraik kasrlarni qo‘shing va ayiring:
Yechilishi:
1)
𝑎−1/
3)
𝑐−1/
1/
𝑎
𝑎 + 𝑎−1 =
𝑐+
𝑎2 −𝑎+𝑎
𝑐−1/
𝑎−1
1/ 2
𝑐
1 − 𝑐−1 =
𝑎2
= 𝑎−1;
2)
𝑐 2 −𝑐+𝑐−1−𝑐 2
𝑐−1
−1
= 𝑐−1;
4)
𝑏−2/
1/
𝑏
𝑏 − 𝑏−2 =
𝑎2
−
𝑎+1
𝑎+1/
𝑏 2 −2𝑏−𝑏
𝑎+
𝑏−2
𝑎+1/
=
1=
𝑏 2 −3𝑏
𝑏−2
;
𝑎2 −𝑎2 −𝑎+𝑎+1
𝑎+1
1
= 𝑎+1.
504-misol. Ifodani soddalashtirib, so‘ngra son qiymatini toping:
Yechilishi:
1)
8𝑎2
𝑎3 −1
(𝑎−1)/
1/
8𝑎2
𝑎+1
+ 𝑎2+𝑎+1, bunda 𝑎 = 2;
+
𝑎3 −1
2)
3𝑐 2 −𝑐+3
𝑐 3 −1
𝑎+1
=
𝑎2 +𝑎+1
𝑐−1
8𝑎2 +𝑎2 −1
𝑎3 −1
2
=
9𝑎2 −1
9𝑎2 −1
,
𝑎3 −1
𝑎3 −1
1
− 𝑐 2 +𝑐+1 + 1−𝑐, bunda 𝑐 = 1 2;
=
9∙22 −1
23 −1
=
9∙4−1
8−1
=
36−1
8−1
=
35
7
= 5.
Javob: 5.
3𝑐 2 −𝑐+3
𝑐 3 −1
=
𝑐−1
2
− 𝑐 2+𝑐+1 + 1−𝑐 =
3𝑐 2 −𝑐+3
3𝑐 2 −𝑐+3−𝑐 2 +2𝑐−1−2𝑐 2 −2𝑐−2
𝑐 3 −1
𝑐
− 𝑐 3 −1 = −
1
2
1 3
(1 ) −1
2
1
=−
(𝑐−1)/
1/
𝑐 3 −1
−
𝑐−1
(𝑐2 +𝑐+1)
−
𝑐 2 +𝑐+1
−𝑐
2
=
𝑐−1
3𝑐 2 −𝑐+3−(𝑐 2 −2𝑐+1)−2(𝑐 2 +𝑐+1)
𝑐 3 −1
𝑐
= 𝑐 3−1 = − 𝑐 3−1;
3
2
3 3
( ) −1
2
3
= − 272
8
−1
3
2
= − 19
=−
8
3
12
∙
84
19
12
= − 19.
12
Javob: − 19.
=
Name:
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