y=xtext{and}7x+3y=10 eching

$$y = x \text { and } 7 x + 3 y = 10$$

y, x uchun yechish

$$x=1$$

$$y=1$$

O'rnini bosishdan foydalanish qoidalari

Birinchi tenglamani yeching. Ikkala tarafdan x ni ayirish.

$$y-x=0$$

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.

$$y-x=0,3y+7x=10$$

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

$$y-x=0$$

x ni tenglamaning ikkala tarafiga qo'shish.

$$y=x$$

x ni y uchun boshqa tenglamada almashtirish, 3y+7x=10.

$$3x+7x=10$$

3x ni 7x ga qo'shish.

$$10x=10$$

Ikki tarafini 10 ga bo‘ling.

$$x=1$$

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

$$y=1$$

Tizim hal qilindi.

$$y=1,x=1$$

Matritsalardan foydalanish qadamlari

Birinchi tenglamani yeching. Ikkala tarafdan x ni ayirish.

$$y-x=0$$

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

$$y-x=0,3y+7x=10$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}1&-1\\3&7\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\10\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-1\\3&7\end{matrix}\right))\left(\begin{matrix}1&-1\\3&7\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\3&7\end{matrix}\right))\left(\begin{matrix}0\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\3&7\end{matrix}\right))\left(\begin{matrix}0\\10\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\3&7\end{matrix}\right))\left(\begin{matrix}0\\10\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{7}{7-\left(-3\right)}&-\frac{-1}{7-\left(-3\right)}\\-\frac{3}{7-\left(-3\right)}&\frac{1}{7-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}0\\10\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{7}{10}&\frac{1}{10}\\-\frac{3}{10}&\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}0\\10\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10}\times 10\\\frac{1}{10}\times 10\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}1\\1\end{matrix}\right)$$

y va x matritsa elementlarini chiqarib olish.

$$y=1,x=1$$

Chiqarib tashlashdan foydalanish qadamlari

Birinchi tenglamani yeching. Ikkala tarafdan x ni ayirish.

$$y-x=0$$

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.

$$y-x=0,3y+7x=10$$

y va 3y ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 3 ga va ikkinchining har bir tarafidagi barcha shartlarni 1 ga ko'paytiring.

$$3y+3\left(-1\right)x=0,3y+7x=10$$

Qisqartirish.

$$3y-3x=0,3y+7x=10$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 3y-3x=0 dan 3y+7x=10 ni ayirish.

$$3y-3y-3x-7x=-10$$

3y ni -3y ga qo'shish. 3y va -3y shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$-3x-7x=-10$$

-3x ni -7x ga qo'shish.

$$-10x=-10$$

Ikki tarafini -10 ga bo‘ling.

$$x=1$$

1 ni x uchun 3y+7x=10 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz y ni bevosita yecha olasiz.

$$3y+7=10$$

Tenglamaning ikkala tarafidan 7 ni ayirish.

$$3y=3$$

Ikki tarafini 3 ga bo‘ling.

$$y=1$$

Tizim hal qilindi.

$$y=1,x=1$$