y=1/3xquadtext{26}y=-5x eching

$$y = \frac { 1 } { 3 } x \quad \text { 26 } y = - 5 x$$

y, x uchun yechish

$$x=0$$

$$y=0$$

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. 5x ni ikki tarafga qo’shing.

$$y+5x=0$$

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.

$$y-\frac{1}{3}x=0,y+5x=0$$

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+5x=0.

$$\frac{1}{3}x+5x=0$$

\frac{x}{3} ni 5x ga qo'shish.

$$\frac{16}{3}x=0$$

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

$$x=0$$

0 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.

$$y=0$$

Tizim hal qilindi.

$$y=0,x=0$$

Matritsalardan foydalanish qadamlari

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

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

Ikkinchi tenglamani yeching. 5x ni ikki tarafga qo’shing.

$$y+5x=0$$

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

$$y-\frac{1}{3}x=0,y+5x=0$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}1&-\frac{1}{3}\\1&5\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\0\end{matrix}\right)$$

\left(\begin{matrix}1&-\frac{1}{3}\\1&5\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}1&-\frac{1}{3}\\1&5\end{matrix}\right))\left(\begin{matrix}1&-\frac{1}{3}\\1&5\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{1}{3}\\1&5\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{1}{3}\\1&5\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{1}{3}\\1&5\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{5}{5-\left(-\frac{1}{3}\right)}&-\frac{-\frac{1}{3}}{5-\left(-\frac{1}{3}\right)}\\-\frac{1}{5-\left(-\frac{1}{3}\right)}&\frac{1}{5-\left(-\frac{1}{3}\right)}\end{matrix}\right)\left(\begin{matrix}0\\0\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{15}{16}&\frac{1}{16}\\-\frac{3}{16}&\frac{3}{16}\end{matrix}\right)\left(\begin{matrix}0\\0\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\0\end{matrix}\right)$$

y va x matritsa elementlarini chiqarib olish.

$$y=0,x=0$$

Chiqarib tashlashdan foydalanish qadamlari

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

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

Ikkinchi tenglamani yeching. 5x ni ikki tarafga qo’shing.

$$y+5x=0$$

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.

$$y-\frac{1}{3}x=0,y+5x=0$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali y-\frac{1}{3}x=0 dan y+5x=0 ni ayirish.

$$y-y-\frac{1}{3}x-5x=0$$

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

$$-\frac{1}{3}x-5x=0$$

-\frac{x}{3} ni -5x ga qo'shish.

$$-\frac{16}{3}x=0$$

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

$$x=0$$

0 ni x uchun y+5x=0 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz y ni bevosita yecha olasiz.

$$y=0$$

Tizim hal qilindi.

$$y=0,x=0$$