y=2x-1text{and}2x+y=3 eching

$$y = 2 x - 1 \text { and } 2 x + y = 3$$

y, x uchun yechish

$$x=1$$

$$y=1$$

O'rnini bosishdan foydalanish qoidalari

Birinchi tenglamani yeching. Ikkala tarafdan 2x ni ayirish.

$$y-2x=-1$$

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.

$$y-2x=-1,y+2x=3$$

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

$$y-2x=-1$$

2x ni tenglamaning ikkala tarafiga qo'shish.

$$y=2x-1$$

2x-1 ni y uchun boshqa tenglamada almashtirish, y+2x=3.

$$2x-1+2x=3$$

2x ni 2x ga qo'shish.

$$4x-1=3$$

1 ni tenglamaning ikkala tarafiga qo'shish.

$$4x=4$$

Ikki tarafini 4 ga bo‘ling.

$$x=1$$

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

$$y=2-1$$

-1 ni 2 ga qo'shish.

$$y=1$$

Tizim hal qilindi.

$$y=1,x=1$$

Matritsalardan foydalanish qadamlari

Birinchi tenglamani yeching. Ikkala tarafdan 2x ni ayirish.

$$y-2x=-1$$

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

$$y-2x=-1,y+2x=3$$

Tenglamalarni matritsa shaklida yozish.

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

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

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

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

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

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

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

\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.

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

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{1}{2}\\-\frac{1}{4}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}-1\\3\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\left(-1\right)+\frac{1}{2}\times 3\\-\frac{1}{4}\left(-1\right)+\frac{1}{4}\times 3\end{matrix}\right)$$

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 2x ni ayirish.

$$y-2x=-1$$

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.

$$y-2x=-1,y+2x=3$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali y-2x=-1 dan y+2x=3 ni ayirish.

$$y-y-2x-2x=-1-3$$

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

$$-2x-2x=-1-3$$

-2x ni -2x ga qo'shish.

$$-4x=-1-3$$

-1 ni -3 ga qo'shish.

$$-4x=-4$$

Ikki tarafini -4 ga bo‘ling.

$$x=1$$

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

$$y+2=3$$

Tenglamaning ikkala tarafidan 2 ni ayirish.

$$y=1$$

Tizim hal qilindi.

$$y=1,x=1$$