-6y+2x=12text{and}x-3(x+2)=2y-8x eching

$$- 6 y + 2 x = 12 \text { and } x - 3 ( x + 2 ) = 2 y - 8 x$$

y, x uchun yechish

$$x=\frac{3}{8}=0,375$$

$$y = -\frac{15}{8} = -1\frac{7}{8} = -1,875$$

O'rnini bosishdan foydalanish qoidalari

Ikkinchi tenglamani yeching. -3 ga x+2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x-3x-6=2y-8x$$

-2x ni olish uchun x va -3x ni birlashtirish.

$$-2x-6=2y-8x$$

Ikkala tarafdan 2y ni ayirish.

$$-2x-6-2y=-8x$$

8x ni ikki tarafga qo’shing.

$$-2x-6-2y+8x=0$$

6x ni olish uchun -2x va 8x ni birlashtirish.

$$6x-6-2y=0$$

6 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$6x-2y=6$$

Almashtirishdan foydalanib tenglamalar juftligini yechish uchun, avval o'zgaruvchan qiymatlardan biri uchun tenglamani yeching. So'ngra ana shu o'zgaruvchan natijani boshqa tenglama bilan almashtiring.

$$-6y+2x=12,-2y+6x=6$$

Tenglamalardan birini tanlang va teng belgisining chap tomonidagi y ni izolyatsiyalash orqali y ni hisoblang.

$$-6y+2x=12$$

Tenglamaning ikkala tarafidan 2x ni ayirish.

$$-6y=-2x+12$$

Ikki tarafini -6 ga bo‘ling.

$$y=-\frac{1}{6}\left(-2x+12\right)$$

-\frac{1}{6} ni -2x+12 marotabaga ko'paytirish.

$$y=\frac{1}{3}x-2$$

\frac{x}{3}-2 ni y uchun boshqa tenglamada almashtirish, -2y+6x=6.

$$-2\left(\frac{1}{3}x-2\right)+6x=6$$

-2 ni \frac{x}{3}-2 marotabaga ko'paytirish.

$$-\frac{2}{3}x+4+6x=6$$

-\frac{2x}{3} ni 6x ga qo'shish.

$$\frac{16}{3}x+4=6$$

Tenglamaning ikkala tarafidan 4 ni ayirish.

$$\frac{16}{3}x=2$$

Tenglamaning ikki tarafini \frac{16}{3} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$x=\frac{3}{8}$$

\frac{3}{8} ni x uchun y=\frac{1}{3}x-2 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz y ni bevosita yecha olasiz.

$$y=\frac{1}{3}\times \frac{3}{8}-2$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{1}{3} ni \frac{3}{8} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$y=\frac{1}{8}-2$$

-2 ni \frac{1}{8} ga qo'shish.

$$y=-\frac{15}{8}$$

Tizim hal qilindi.

$$y=-\frac{15}{8},x=\frac{3}{8}$$

Matritsalardan foydalanish qadamlari

Ikkinchi tenglamani yeching. -3 ga x+2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x-3x-6=2y-8x$$

-2x ni olish uchun x va -3x ni birlashtirish.

$$-2x-6=2y-8x$$

Ikkala tarafdan 2y ni ayirish.

$$-2x-6-2y=-8x$$

8x ni ikki tarafga qo’shing.

$$-2x-6-2y+8x=0$$

6x ni olish uchun -2x va 8x ni birlashtirish.

$$6x-6-2y=0$$

6 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$6x-2y=6$$

Tenglamalar standart shaklda ko'rsatilsin so'ng tenglamalar tizimini yechish uchun matritsalardan foydalanilsin.

$$-6y+2x=12,-2y+6x=6$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}-6&2\\-2&6\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}12\\6\end{matrix}\right)$$

\left(\begin{matrix}-6&2\\-2&6\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}-6&2\\-2&6\end{matrix}\right))\left(\begin{matrix}-6&2\\-2&6\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}-6&2\\-2&6\end{matrix}\right))\left(\begin{matrix}12\\6\end{matrix}\right)$$

Matritsaning ko‘paytmasi va teskarisi o‘zaro teng matristsadir.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}-6&2\\-2&6\end{matrix}\right))\left(\begin{matrix}12\\6\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}-6&2\\-2&6\end{matrix}\right))\left(\begin{matrix}12\\6\end{matrix}\right)$$

\left(\begin{matrix}a&b\\c&d\end{matrix}\right) 2\times 2 matrix uchun, teskari matritsa \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), shuning uchun matritsa tenglamasini matritsani ko‘paytirish masalasi sifatida qayta yozish mumkin.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{6}{-6\times 6-2\left(-2\right)}&-\frac{2}{-6\times 6-2\left(-2\right)}\\-\frac{-2}{-6\times 6-2\left(-2\right)}&-\frac{6}{-6\times 6-2\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}12\\6\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{16}&\frac{1}{16}\\-\frac{1}{16}&\frac{3}{16}\end{matrix}\right)\left(\begin{matrix}12\\6\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{16}\times 12+\frac{1}{16}\times 6\\-\frac{1}{16}\times 12+\frac{3}{16}\times 6\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{15}{8}\\\frac{3}{8}\end{matrix}\right)$$

y va x matritsa elementlarini chiqarib olish.

$$y=-\frac{15}{8},x=\frac{3}{8}$$

Chiqarib tashlashdan foydalanish qadamlari

Ikkinchi tenglamani yeching. -3 ga x+2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x-3x-6=2y-8x$$

-2x ni olish uchun x va -3x ni birlashtirish.

$$-2x-6=2y-8x$$

Ikkala tarafdan 2y ni ayirish.

$$-2x-6-2y=-8x$$

8x ni ikki tarafga qo’shing.

$$-2x-6-2y+8x=0$$

6x ni olish uchun -2x va 8x ni birlashtirish.

$$6x-6-2y=0$$

6 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$6x-2y=6$$

Chiqarib tashlash bilan yechim hosil qilish uchun, o'zgartmalarning koeffitsienti ikkala tenglamada bir xil bo'lib o'zgaruvchan qiymat birining boshqasidan ayirilganda, bekor qilishi lozim.

$$-6y+2x=12,-2y+6x=6$$

-6y va -2y ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni -2 ga va ikkinchining har bir tarafidagi barcha shartlarni -6 ga ko'paytiring.

$$-2\left(-6\right)y-2\times 2x=-2\times 12,-6\left(-2\right)y-6\times 6x=-6\times 6$$

Qisqartirish.

$$12y-4x=-24,12y-36x=-36$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 12y-4x=-24 dan 12y-36x=-36 ni ayirish.

$$12y-12y-4x+36x=-24+36$$

12y ni -12y ga qo'shish. 12y va -12y shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$-4x+36x=-24+36$$

-4x ni 36x ga qo'shish.

$$32x=-24+36$$

-24 ni 36 ga qo'shish.

$$32x=12$$

Ikki tarafini 32 ga bo‘ling.

$$x=\frac{3}{8}$$

\frac{3}{8} ni x uchun -2y+6x=6 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz y ni bevosita yecha olasiz.

$$-2y+6\times \frac{3}{8}=6$$

6 ni \frac{3}{8} marotabaga ko'paytirish.

$$-2y+\frac{9}{4}=6$$

Tenglamaning ikkala tarafidan \frac{9}{4} ni ayirish.

$$-2y=\frac{15}{4}$$

Ikki tarafini -2 ga bo‘ling.

$$y=-\frac{15}{8}$$

Tizim hal qilindi.

$$y=-\frac{15}{8},x=\frac{3}{8}$$