x-1/x-2+x-7/x-8=x-3/x-4+x-5/x-6 eching

$$\frac{ x-1 }{ x-2 } + \frac{ x-7 }{ x-8 } = \frac{ x-3 }{ x-4 } + \frac{ x-5 }{ x-6 }$$

x uchun yechish

$$x=5$$

Chiziqli tenglamalarni yechish bo'yicha qadamlar

x qiymati 2,4,6,8 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-8\right)\left(x-6\right)\left(x-4\right)\left(x-2\right) ga, x-2,x-8,x-4,x-6 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-8\right)\left(x-6\right)\left(x-4\right)\left(x-1\right)+\left(x-6\right)\left(x-4\right)\left(x-2\right)\left(x-7\right)=\left(x-8\right)\left(x-6\right)\left(x-2\right)\left(x-3\right)+\left(x-8\right)\left(x-4\right)\left(x-2\right)\left(x-5\right)$$

x-8 ga x-6 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$\left(x^{2}-14x+48\right)\left(x-4\right)\left(x-1\right)+\left(x-6\right)\left(x-4\right)\left(x-2\right)\left(x-7\right)=\left(x-8\right)\left(x-6\right)\left(x-2\right)\left(x-3\right)+\left(x-8\right)\left(x-4\right)\left(x-2\right)\left(x-5\right)$$

x^{2}-14x+48 ga x-4 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$\left(x^{3}-18x^{2}+104x-192\right)\left(x-1\right)+\left(x-6\right)\left(x-4\right)\left(x-2\right)\left(x-7\right)=\left(x-8\right)\left(x-6\right)\left(x-2\right)\left(x-3\right)+\left(x-8\right)\left(x-4\right)\left(x-2\right)\left(x-5\right)$$

x^{3}-18x^{2}+104x-192 ga x-1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{4}-19x^{3}+122x^{2}-296x+192+\left(x-6\right)\left(x-4\right)\left(x-2\right)\left(x-7\right)=\left(x-8\right)\left(x-6\right)\left(x-2\right)\left(x-3\right)+\left(x-8\right)\left(x-4\right)\left(x-2\right)\left(x-5\right)$$

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

$$x^{4}-19x^{3}+122x^{2}-296x+192+\left(x^{2}-10x+24\right)\left(x-2\right)\left(x-7\right)=\left(x-8\right)\left(x-6\right)\left(x-2\right)\left(x-3\right)+\left(x-8\right)\left(x-4\right)\left(x-2\right)\left(x-5\right)$$

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

$$x^{4}-19x^{3}+122x^{2}-296x+192+\left(x^{3}-12x^{2}+44x-48\right)\left(x-7\right)=\left(x-8\right)\left(x-6\right)\left(x-2\right)\left(x-3\right)+\left(x-8\right)\left(x-4\right)\left(x-2\right)\left(x-5\right)$$

x^{3}-12x^{2}+44x-48 ga x-7 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{4}-19x^{3}+122x^{2}-296x+192+x^{4}-19x^{3}+128x^{2}-356x+336=\left(x-8\right)\left(x-6\right)\left(x-2\right)\left(x-3\right)+\left(x-8\right)\left(x-4\right)\left(x-2\right)\left(x-5\right)$$

2x^{4} ni olish uchun x^{4} va x^{4} ni birlashtirish.

$$2x^{4}-19x^{3}+122x^{2}-296x+192-19x^{3}+128x^{2}-356x+336=\left(x-8\right)\left(x-6\right)\left(x-2\right)\left(x-3\right)+\left(x-8\right)\left(x-4\right)\left(x-2\right)\left(x-5\right)$$

-38x^{3} ni olish uchun -19x^{3} va -19x^{3} ni birlashtirish.

$$2x^{4}-38x^{3}+122x^{2}-296x+192+128x^{2}-356x+336=\left(x-8\right)\left(x-6\right)\left(x-2\right)\left(x-3\right)+\left(x-8\right)\left(x-4\right)\left(x-2\right)\left(x-5\right)$$

250x^{2} ni olish uchun 122x^{2} va 128x^{2} ni birlashtirish.

$$2x^{4}-38x^{3}+250x^{2}-296x+192-356x+336=\left(x-8\right)\left(x-6\right)\left(x-2\right)\left(x-3\right)+\left(x-8\right)\left(x-4\right)\left(x-2\right)\left(x-5\right)$$

-652x ni olish uchun -296x va -356x ni birlashtirish.

$$2x^{4}-38x^{3}+250x^{2}-652x+192+336=\left(x-8\right)\left(x-6\right)\left(x-2\right)\left(x-3\right)+\left(x-8\right)\left(x-4\right)\left(x-2\right)\left(x-5\right)$$

528 olish uchun 192 va 336'ni qo'shing.

$$2x^{4}-38x^{3}+250x^{2}-652x+528=\left(x-8\right)\left(x-6\right)\left(x-2\right)\left(x-3\right)+\left(x-8\right)\left(x-4\right)\left(x-2\right)\left(x-5\right)$$

x-8 ga x-6 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{4}-38x^{3}+250x^{2}-652x+528=\left(x^{2}-14x+48\right)\left(x-2\right)\left(x-3\right)+\left(x-8\right)\left(x-4\right)\left(x-2\right)\left(x-5\right)$$

x^{2}-14x+48 ga x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{4}-38x^{3}+250x^{2}-652x+528=\left(x^{3}-16x^{2}+76x-96\right)\left(x-3\right)+\left(x-8\right)\left(x-4\right)\left(x-2\right)\left(x-5\right)$$

x^{3}-16x^{2}+76x-96 ga x-3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{4}-38x^{3}+250x^{2}-652x+528=x^{4}-19x^{3}+124x^{2}-324x+288+\left(x-8\right)\left(x-4\right)\left(x-2\right)\left(x-5\right)$$

x-8 ga x-4 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{4}-38x^{3}+250x^{2}-652x+528=x^{4}-19x^{3}+124x^{2}-324x+288+\left(x^{2}-12x+32\right)\left(x-2\right)\left(x-5\right)$$

x^{2}-12x+32 ga x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{4}-38x^{3}+250x^{2}-652x+528=x^{4}-19x^{3}+124x^{2}-324x+288+\left(x^{3}-14x^{2}+56x-64\right)\left(x-5\right)$$

x^{3}-14x^{2}+56x-64 ga x-5 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{4}-38x^{3}+250x^{2}-652x+528=x^{4}-19x^{3}+124x^{2}-324x+288+x^{4}-19x^{3}+126x^{2}-344x+320$$

2x^{4} ni olish uchun x^{4} va x^{4} ni birlashtirish.

$$2x^{4}-38x^{3}+250x^{2}-652x+528=2x^{4}-19x^{3}+124x^{2}-324x+288-19x^{3}+126x^{2}-344x+320$$

-38x^{3} ni olish uchun -19x^{3} va -19x^{3} ni birlashtirish.

$$2x^{4}-38x^{3}+250x^{2}-652x+528=2x^{4}-38x^{3}+124x^{2}-324x+288+126x^{2}-344x+320$$

250x^{2} ni olish uchun 124x^{2} va 126x^{2} ni birlashtirish.

$$2x^{4}-38x^{3}+250x^{2}-652x+528=2x^{4}-38x^{3}+250x^{2}-324x+288-344x+320$$

-668x ni olish uchun -324x va -344x ni birlashtirish.

$$2x^{4}-38x^{3}+250x^{2}-652x+528=2x^{4}-38x^{3}+250x^{2}-668x+288+320$$

608 olish uchun 288 va 320'ni qo'shing.

$$2x^{4}-38x^{3}+250x^{2}-652x+528=2x^{4}-38x^{3}+250x^{2}-668x+608$$

Ikkala tarafdan 2x^{4} ni ayirish.

$$2x^{4}-38x^{3}+250x^{2}-652x+528-2x^{4}=-38x^{3}+250x^{2}-668x+608$$

0 ni olish uchun 2x^{4} va -2x^{4} ni birlashtirish.

$$-38x^{3}+250x^{2}-652x+528=-38x^{3}+250x^{2}-668x+608$$

38x^{3} ni ikki tarafga qo’shing.

$$-38x^{3}+250x^{2}-652x+528+38x^{3}=250x^{2}-668x+608$$

0 ni olish uchun -38x^{3} va 38x^{3} ni birlashtirish.

$$250x^{2}-652x+528=250x^{2}-668x+608$$

Ikkala tarafdan 250x^{2} ni ayirish.

$$250x^{2}-652x+528-250x^{2}=-668x+608$$

0 ni olish uchun 250x^{2} va -250x^{2} ni birlashtirish.

$$-652x+528=-668x+608$$

668x ni ikki tarafga qo’shing.

$$-652x+528+668x=608$$

16x ni olish uchun -652x va 668x ni birlashtirish.

$$16x+528=608$$

Ikkala tarafdan 528 ni ayirish.

$$16x=608-528$$

80 olish uchun 608 dan 528 ni ayirish.

$$16x=80$$

Ikki tarafini 16 ga bo‘ling.

$$x=\frac{80}{16}$$

5 ni olish uchun 80 ni 16 ga bo‘ling.

$$x=5$$