y=x-9;quady=5-x eching

$$y = x - 9 ; \quad y = 5 - x$$

y, x uchun yechish

$$x=7$$

$$y=-2$$

O'rnini bosishdan foydalanish qoidalari

Birinchi tenglamani yeching. Ikkala tarafdan x ni ayirish.

$$y-x=-9$$

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

$$y+x=5$$

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.

$$y-x=-9,y+x=5$$

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

$$y-x=-9$$

x ni tenglamaning ikkala tarafiga qo'shish.

$$y=x-9$$

x-9 ni y uchun boshqa tenglamada almashtirish, y+x=5.

$$x-9+x=5$$

x ni x ga qo'shish.

$$2x-9=5$$

9 ni tenglamaning ikkala tarafiga qo'shish.

$$2x=14$$

Ikki tarafini 2 ga bo‘ling.

$$x=7$$

7 ni x uchun y=x-9 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz y ni bevosita yecha olasiz.

$$y=7-9$$

-9 ni 7 ga qo'shish.

$$y=-2$$

Tizim hal qilindi.

$$y=-2,x=7$$

Matritsalardan foydalanish qadamlari

Birinchi tenglamani yeching. Ikkala tarafdan x ni ayirish.

$$y-x=-9$$

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

$$y+x=5$$

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

$$y-x=-9,y+x=5$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}1&-1\\1&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-9\\5\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-1\\1&1\end{matrix}\right))\left(\begin{matrix}1&-1\\1&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&1\end{matrix}\right))\left(\begin{matrix}-9\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&1\end{matrix}\right))\left(\begin{matrix}-9\\5\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&1\end{matrix}\right))\left(\begin{matrix}-9\\5\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-\left(-1\right)}&-\frac{-1}{1-\left(-1\right)}\\-\frac{1}{1-\left(-1\right)}&\frac{1}{1-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}-9\\5\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{1}{2}\\-\frac{1}{2}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}-9\\5\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\left(-9\right)+\frac{1}{2}\times 5\\-\frac{1}{2}\left(-9\right)+\frac{1}{2}\times 5\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

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

y va x matritsa elementlarini chiqarib olish.

$$y=-2,x=7$$

Chiqarib tashlashdan foydalanish qadamlari

Birinchi tenglamani yeching. Ikkala tarafdan x ni ayirish.

$$y-x=-9$$

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

$$y+x=5$$

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.

$$y-x=-9,y+x=5$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali y-x=-9 dan y+x=5 ni ayirish.

$$y-y-x-x=-9-5$$

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

$$-x-x=-9-5$$

-x ni -x ga qo'shish.

$$-2x=-9-5$$

-9 ni -5 ga qo'shish.

$$-2x=-14$$

Ikki tarafini -2 ga bo‘ling.

$$x=7$$

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

$$y+7=5$$

Tenglamaning ikkala tarafidan 7 ni ayirish.

$$y=-2$$

Tizim hal qilindi.

$$y=-2,x=7$$