1/x-3-1/x-4=1/x-5-1/x-6 eching

$$\frac { 1 } { x - 3 } - \frac { 1 } { x - 4 } = \frac { 1 } { x - 5 } - \frac { 1 } { x - 6 }$$

x uchun yechish

$$x = \frac{9}{2} = 4\frac{1}{2} = 4,5$$

Chiziqli tenglamalarni yechish bo'yicha qadamlar

x qiymati 3,4,5,6 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-6\right)\left(x-5\right)\left(x-4\right)\left(x-3\right) ga, x-3,x-4,x-5,x-6 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-6\right)\left(x-5\right)\left(x-4\right)-\left(x-6\right)\left(x-5\right)\left(x-3\right)=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

x-6 ga x-5 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$\left(x^{2}-11x+30\right)\left(x-4\right)-\left(x-6\right)\left(x-5\right)\left(x-3\right)=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

x^{2}-11x+30 ga x-4 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{3}-15x^{2}+74x-120-\left(x-6\right)\left(x-5\right)\left(x-3\right)=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

x-6 ga x-5 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{3}-15x^{2}+74x-120-\left(x^{2}-11x+30\right)\left(x-3\right)=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

x^{2}-11x+30 ga x-3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{3}-15x^{2}+74x-120-\left(x^{3}-14x^{2}+63x-90\right)=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

x^{3}-14x^{2}+63x-90 teskarisini topish uchun har birining teskarisini toping.

$$x^{3}-15x^{2}+74x-120-x^{3}+14x^{2}-63x+90=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

0 ni olish uchun x^{3} va -x^{3} ni birlashtirish.

$$-15x^{2}+74x-120+14x^{2}-63x+90=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

-x^{2} ni olish uchun -15x^{2} va 14x^{2} ni birlashtirish.

$$-x^{2}+74x-120-63x+90=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

11x ni olish uchun 74x va -63x ni birlashtirish.

$$-x^{2}+11x-120+90=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

-30 olish uchun -120 va 90'ni qo'shing.

$$-x^{2}+11x-30=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

x-6 ga x-4 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-x^{2}+11x-30=\left(x^{2}-10x+24\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

x^{2}-10x+24 ga x-3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-x^{2}+11x-30=x^{3}-13x^{2}+54x-72-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

x-5 ga x-4 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-x^{2}+11x-30=x^{3}-13x^{2}+54x-72-\left(x^{2}-9x+20\right)\left(x-3\right)$$

x^{2}-9x+20 ga x-3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-x^{2}+11x-30=x^{3}-13x^{2}+54x-72-\left(x^{3}-12x^{2}+47x-60\right)$$

x^{3}-12x^{2}+47x-60 teskarisini topish uchun har birining teskarisini toping.

$$-x^{2}+11x-30=x^{3}-13x^{2}+54x-72-x^{3}+12x^{2}-47x+60$$

0 ni olish uchun x^{3} va -x^{3} ni birlashtirish.

$$-x^{2}+11x-30=-13x^{2}+54x-72+12x^{2}-47x+60$$

-x^{2} ni olish uchun -13x^{2} va 12x^{2} ni birlashtirish.

$$-x^{2}+11x-30=-x^{2}+54x-72-47x+60$$

7x ni olish uchun 54x va -47x ni birlashtirish.

$$-x^{2}+11x-30=-x^{2}+7x-72+60$$

-12 olish uchun -72 va 60'ni qo'shing.

$$-x^{2}+11x-30=-x^{2}+7x-12$$

x^{2} ni ikki tarafga qo’shing.

$$-x^{2}+11x-30+x^{2}=7x-12$$

0 ni olish uchun -x^{2} va x^{2} ni birlashtirish.

$$11x-30=7x-12$$

Ikkala tarafdan 7x ni ayirish.

$$11x-30-7x=-12$$

4x ni olish uchun 11x va -7x ni birlashtirish.

$$4x-30=-12$$

30 ni ikki tarafga qo’shing.

$$4x=-12+30$$

18 olish uchun -12 va 30'ni qo'shing.

$$4x=18$$

Ikki tarafini 4 ga bo‘ling.

$$x=\frac{18}{4}$$

\frac{18}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{9}{2}$$