y-3x=2text{and}7x-2y=8 eching

$$y - 3 x = 2 \text { and } 7 x - 2 y = 8$$

y, x uchun yechish

$$x=12$$

$$y=38$$

O'rnini bosishdan foydalanish qoidalari

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.

$$y-3x=2,-2y+7x=8$$

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

$$y-3x=2$$

3x ni tenglamaning ikkala tarafiga qo'shish.

$$y=3x+2$$

3x+2 ni y uchun boshqa tenglamada almashtirish, -2y+7x=8.

$$-2\left(3x+2\right)+7x=8$$

-2 ni 3x+2 marotabaga ko'paytirish.

$$-6x-4+7x=8$$

-6x ni 7x ga qo'shish.

$$x-4=8$$

4 ni tenglamaning ikkala tarafiga qo'shish.

$$x=12$$

12 ni x uchun y=3x+2 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz y ni bevosita yecha olasiz.

$$y=3\times 12+2$$

3 ni 12 marotabaga ko'paytirish.

$$y=36+2$$

2 ni 36 ga qo'shish.

$$y=38$$

Tizim hal qilindi.

$$y=38,x=12$$

Matritsalardan foydalanish qadamlari

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

$$y-3x=2,-2y+7x=8$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}1&-3\\-2&7\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}2\\8\end{matrix}\right)$$

\left(\begin{matrix}1&-3\\-2&7\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}1&-3\\-2&7\end{matrix}\right))\left(\begin{matrix}1&-3\\-2&7\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\-2&7\end{matrix}\right))\left(\begin{matrix}2\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\-2&7\end{matrix}\right))\left(\begin{matrix}2\\8\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\-2&7\end{matrix}\right))\left(\begin{matrix}2\\8\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{7}{7-\left(-3\left(-2\right)\right)}&-\frac{-3}{7-\left(-3\left(-2\right)\right)}\\-\frac{-2}{7-\left(-3\left(-2\right)\right)}&\frac{1}{7-\left(-3\left(-2\right)\right)}\end{matrix}\right)\left(\begin{matrix}2\\8\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}7&3\\2&1\end{matrix}\right)\left(\begin{matrix}2\\8\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}7\times 2+3\times 8\\2\times 2+8\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}38\\12\end{matrix}\right)$$

y va x matritsa elementlarini chiqarib olish.

$$y=38,x=12$$

Chiqarib tashlashdan foydalanish qadamlari

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.

$$y-3x=2,-2y+7x=8$$

y va -2y ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni -2 ga va ikkinchining har bir tarafidagi barcha shartlarni 1 ga ko'paytiring.

$$-2y-2\left(-3\right)x=-2\times 2,-2y+7x=8$$

Qisqartirish.

$$-2y+6x=-4,-2y+7x=8$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali -2y+6x=-4 dan -2y+7x=8 ni ayirish.

$$-2y+2y+6x-7x=-4-8$$

-2y ni 2y ga qo'shish. -2y va 2y shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$6x-7x=-4-8$$

6x ni -7x ga qo'shish.

$$-x=-4-8$$

-4 ni -8 ga qo'shish.

$$-x=-12$$

Ikki tarafini -1 ga bo‘ling.

$$x=12$$

12 ni x uchun -2y+7x=8 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz y ni bevosita yecha olasiz.

$$-2y+7\times 12=8$$

7 ni 12 marotabaga ko'paytirish.

$$-2y+84=8$$

Tenglamaning ikkala tarafidan 84 ni ayirish.

$$-2y=-76$$

Ikki tarafini -2 ga bo‘ling.

$$y=38$$

Tizim hal qilindi.

$$y=38,x=12$$