(3k+1)x^2+3k-1/(3k+1)(3k^2+8k-3)+x/9k^2-1=1/3k^2+10k+3 eching

$$\frac { ( 3 k + 1 ) x ^ { 2 } + 3 k - 1 } { ( 3 k + 1 ) ( 3 k ^ { 2 } + 8 k - 3 ) } + \frac { x } { 9 k ^ { 2 } - 1 } = \frac { 1 } { 3 k ^ { 2 } + 10 k + 3 }$$

k uchun yechish (complex solution)

$$\left\{\begin{matrix}k=-\frac{x+3}{3x+1}\text{, }&x\neq -\frac{1}{3}\text{ and }x\neq 0\text{ and }x\neq -\frac{5}{3}\\k\in \mathrm{C}\setminus -\frac{1}{3},-3,\frac{1}{3}\text{, }&x=0\end{matrix}\right,$$

Chiziqli tenglamalarni yechish bo'yicha qadamlar

k qiymati -3,-\frac{1}{3},\frac{1}{3} qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(3k-1\right)\left(k+3\right)\left(3k+1\right) ga, \left(3k+1\right)\left(3k^{2}+8k-3\right),9k^{2}-1,3k^{2}+10k+3 ning eng kichik karralisiga ko‘paytiring.

$$\left(3k+1\right)x^{2}+3k-1+\left(k+3\right)x=3k-1$$

3k+1 ga x^{2} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3kx^{2}+x^{2}+3k-1+\left(k+3\right)x=3k-1$$

k+3 ga x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3kx^{2}+x^{2}+3k-1+kx+3x=3k-1$$

Ikkala tarafdan 3k ni ayirish.

$$3kx^{2}+x^{2}+3k-1+kx+3x-3k=-1$$

0 ni olish uchun 3k va -3k ni birlashtirish.

$$3kx^{2}+x^{2}-1+kx+3x=-1$$

Ikkala tarafdan x^{2} ni ayirish.

$$3kx^{2}-1+kx+3x=-1-x^{2}$$

1 ni ikki tarafga qo’shing.

$$3kx^{2}+kx+3x=-1-x^{2}+1$$

0 olish uchun -1 va 1'ni qo'shing.

$$3kx^{2}+kx+3x=-x^{2}$$

Ikkala tarafdan 3x ni ayirish.

$$3kx^{2}+kx=-x^{2}-3x$$

k'ga ega bo'lgan barcha shartlarni birlashtirish.

$$\left(3x^{2}+x\right)k=-x^{2}-3x$$

Ikki tarafini 3x^{2}+x ga bo‘ling.

$$\frac{\left(3x^{2}+x\right)k}{3x^{2}+x}=-\frac{x\left(x+3\right)}{3x^{2}+x}$$

3x^{2}+x ga bo'lish 3x^{2}+x ga ko'paytirishni bekor qiladi.

$$k=-\frac{x\left(x+3\right)}{3x^{2}+x}$$

-x\left(3+x\right) ni 3x^{2}+x ga bo'lish.

$$k=-\frac{x+3}{3x+1}$$

k qiymati -\frac{1}{3},-3,\frac{1}{3} qiymatlaridan birortasiga teng bo‘lmaydi.

$$k=-\frac{x+3}{3x+1}\text{, }k\neq -\frac{1}{3}\text{ and }k\neq -3\text{ and }k\neq \frac{1}{3}$$

k uchun yechish

$$\left\{\begin{matrix}k=-\frac{x+3}{3x+1}\text{, }&x\neq -\frac{1}{3}\text{ and }x\neq -\frac{5}{3}\text{ and }x\neq 0\\k\in \mathrm{R}\setminus -\frac{1}{3},\frac{1}{3},-3\text{, }&x=0\end{matrix}\right,$$

Chiziqli tenglamalarni yechish bo'yicha qadamlar

k qiymati -3,-\frac{1}{3},\frac{1}{3} qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(3k-1\right)\left(k+3\right)\left(3k+1\right) ga, \left(3k+1\right)\left(3k^{2}+8k-3\right),9k^{2}-1,3k^{2}+10k+3 ning eng kichik karralisiga ko‘paytiring.

$$\left(3k+1\right)x^{2}+3k-1+\left(k+3\right)x=3k-1$$

3k+1 ga x^{2} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3kx^{2}+x^{2}+3k-1+\left(k+3\right)x=3k-1$$

k+3 ga x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3kx^{2}+x^{2}+3k-1+kx+3x=3k-1$$

Ikkala tarafdan 3k ni ayirish.

$$3kx^{2}+x^{2}+3k-1+kx+3x-3k=-1$$

0 ni olish uchun 3k va -3k ni birlashtirish.

$$3kx^{2}+x^{2}-1+kx+3x=-1$$

Ikkala tarafdan x^{2} ni ayirish.

$$3kx^{2}-1+kx+3x=-1-x^{2}$$

1 ni ikki tarafga qo’shing.

$$3kx^{2}+kx+3x=-1-x^{2}+1$$

0 olish uchun -1 va 1'ni qo'shing.

$$3kx^{2}+kx+3x=-x^{2}$$

Ikkala tarafdan 3x ni ayirish.

$$3kx^{2}+kx=-x^{2}-3x$$

k'ga ega bo'lgan barcha shartlarni birlashtirish.

$$\left(3x^{2}+x\right)k=-x^{2}-3x$$

Ikki tarafini 3x^{2}+x ga bo‘ling.

$$\frac{\left(3x^{2}+x\right)k}{3x^{2}+x}=-\frac{x\left(x+3\right)}{3x^{2}+x}$$

3x^{2}+x ga bo'lish 3x^{2}+x ga ko'paytirishni bekor qiladi.

$$k=-\frac{x\left(x+3\right)}{3x^{2}+x}$$

-x\left(3+x\right) ni 3x^{2}+x ga bo'lish.

$$k=-\frac{x+3}{3x+1}$$

k qiymati -\frac{1}{3},-3,\frac{1}{3} qiymatlaridan birortasiga teng bo‘lmaydi.

$$k=-\frac{x+3}{3x+1}\text{, }k\neq -\frac{1}{3}\text{ and }k\neq -3\text{ and }k\neq \frac{1}{3}$$

x uchun yechish (complex solution)

$$x=-\frac{k+3}{3k+1}$$

$$x=0\text{, }k\neq -\frac{1}{3}\text{ and }k\neq -3\text{ and }k\neq \frac{1}{3}$$

x uchun yechish

$$x=-\frac{k+3}{3k+1}$$

$$x=0\text{, }k\neq -3\text{ and }|k|\neq \frac{1}{3}$$