y=1/3x;quady=60-3x eching

$$y = \frac { 1 } { 3 } x ; \quad y = 60 - 3 x$$

y, x uchun yechish

$$x=18$$

$$y=6$$

O'rnini bosishdan foydalanish qoidalari

Birinchi tenglamani yeching. Ikkala tarafdan \frac{1}{3}x ni ayirish.

$$y-\frac{1}{3}x=0$$

Ikkinchi tenglamani yeching. 3x ni ikki tarafga qo’shing.

$$y+3x=60$$

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.

$$y-\frac{1}{3}x=0,y+3x=60$$

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

$$y-\frac{1}{3}x=0$$

\frac{x}{3} ni tenglamaning ikkala tarafiga qo'shish.

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

\frac{x}{3} ni y uchun boshqa tenglamada almashtirish, y+3x=60.

$$\frac{1}{3}x+3x=60$$

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

$$\frac{10}{3}x=60$$

Tenglamaning ikki tarafini \frac{10}{3} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$x=18$$

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

$$y=\frac{1}{3}\times 18$$

\frac{1}{3} ni 18 marotabaga ko'paytirish.

$$y=6$$

Tizim hal qilindi.

$$y=6,x=18$$

Matritsalardan foydalanish qadamlari

Birinchi tenglamani yeching. Ikkala tarafdan \frac{1}{3}x ni ayirish.

$$y-\frac{1}{3}x=0$$

Ikkinchi tenglamani yeching. 3x ni ikki tarafga qo’shing.

$$y+3x=60$$

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

$$y-\frac{1}{3}x=0,y+3x=60$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}1&-\frac{1}{3}\\1&3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\60\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-\frac{1}{3}\\1&3\end{matrix}\right))\left(\begin{matrix}1&-\frac{1}{3}\\1&3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{1}{3}\\1&3\end{matrix}\right))\left(\begin{matrix}0\\60\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{1}{3}\\1&3\end{matrix}\right))\left(\begin{matrix}0\\60\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{1}{3}\\1&3\end{matrix}\right))\left(\begin{matrix}0\\60\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3-\left(-\frac{1}{3}\right)}&-\frac{-\frac{1}{3}}{3-\left(-\frac{1}{3}\right)}\\-\frac{1}{3-\left(-\frac{1}{3}\right)}&\frac{1}{3-\left(-\frac{1}{3}\right)}\end{matrix}\right)\left(\begin{matrix}0\\60\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{9}{10}&\frac{1}{10}\\-\frac{3}{10}&\frac{3}{10}\end{matrix}\right)\left(\begin{matrix}0\\60\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10}\times 60\\\frac{3}{10}\times 60\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}6\\18\end{matrix}\right)$$

y va x matritsa elementlarini chiqarib olish.

$$y=6,x=18$$

Chiqarib tashlashdan foydalanish qadamlari

Birinchi tenglamani yeching. Ikkala tarafdan \frac{1}{3}x ni ayirish.

$$y-\frac{1}{3}x=0$$

Ikkinchi tenglamani yeching. 3x ni ikki tarafga qo’shing.

$$y+3x=60$$

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.

$$y-\frac{1}{3}x=0,y+3x=60$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali y-\frac{1}{3}x=0 dan y+3x=60 ni ayirish.

$$y-y-\frac{1}{3}x-3x=-60$$

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

$$-\frac{1}{3}x-3x=-60$$

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

$$-\frac{10}{3}x=-60$$

Tenglamaning ikki tarafini -\frac{10}{3} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$x=18$$

18 ni x uchun y+3x=60 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz y ni bevosita yecha olasiz.

$$y+3\times 18=60$$

3 ni 18 marotabaga ko'paytirish.

$$y+54=60$$

Tenglamaning ikkala tarafidan 54 ni ayirish.

$$y=6$$

Tizim hal qilindi.

$$y=6,x=18$$